(written from a Production point of view)
In the earliest scripts of Star Trek II: The Wrath of Khan the USS Reliant was supposed to be a Constitution-class. However, the cumbersomeness of the Enterprise studio model from Star Trek: The Motion Picture (vehemently expressed by Special Visual Effects Supervisor Kenneth Ralston at the time) was part of the reasons why the class of the Reliant was redesigned from a Constitution-class to the Miranda-class, the other reason being that the producers were afraid that audiences would not be able to tell the two ships apart during the battle sequences. "In the dogfight you had to instantly recognize which ship you were looking at, so they had to look different. At the same time, you had to make them look like they came from the same culture and had the same technology", Joe Jennings elaborated. (Star Trek: The Magazine Volume 3, Issue 5, p. 69) In the process, the Miranda-class became the first new true Starfleet starship design ever seen by the public in a live action production.
The script called for an older cruiser of "Enterprise's own class" but of a "different configuration." This ended up being a different class of starship in its own right, the first distinct Federation starship design other than the NCC-1701 to ever appear in a filmed, live-action Star Trek production. What was to become the Miranda-class studio model, specifically the USS Reliant, was designed at Paramount Pictures by Joe Jennings, Mike Minor, and Graphic Designer Lee Cole. On the design Jennings noted, "It was the first time a new spaceship had been designed since the Klingon battle cruiser. That was a lot of fun. She was supposed to be a coastal and geodetic survey ship, like a buoy tender. She would be armed perhaps, but only lightly; she wasn't a lion ship like the Enterprise. Also, remember, the Enterprise was supposed to be an exploratory vessel, where the armament was secondary. That was even more true for the Reliant; she was supposed to just stick around in the known universe and take care of things that everybody already knew about. (...) It was fun to try to make it look identifiable different; we had long postulated that the circular saucer said "This is Starfleet Navy," and it uses engines that looked pretty much like those on the Enterprise." (Star Trek: The Magazine Volume 3, Issue 5, pp. 69-70)
The Miranda-class model was originally designed in a reversed orientation, with the nacelles above the dish and two counterbalancing torpedo launch pods below the dish. Lee Cole, however, later explained, that when the initial drawing was sent to Harve Bennett for approval, he viewed the drawings upside down. He liked the design and signed off on it. The visual effects team debated whether or not to send the sketches back to Bennett (who was in Israel at the time working on another project) with the correct orientation, but they decided that there was insufficient time to do so and decided to add a "rollbar" to balance out the suddenly dropped nacelles. (Star Trek: The Magazine Volume 3, Issue 5, p. 70)
A design patent, No. D272839, was issued by the US Patent and Trademark Office to Paramount Pictures Corporation for the design of the USS Reliant on 28 February 1984 (there called a "toy spaceship"), which noted both Joseph R. Jennings and Michael Minor as the "sole inventors" of the design. The patent issued was valid for fourteen years. The patent application was tendered by the studio on 7 June 1982.
A set of "STARSHIP RELIANT MODEL BLUEPRINTS AND STAR TREK II: THE WRATH OF KHAN DEVELOPMENT DOCUMENTS", used in developing the Reliant miniature, and estimated at US$800-$1,200, was sold on 7 October 2006 as Lot 726 in Christie's 40 Years of Star Trek: The Collection auction for US$4,000 ($4,800 with premium).
Large studio model Edit
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The studio model was constructed at Industrial Light & Magic (ILM) under Supervising Modelmaker Steve Gawley. Among Gawley's crew were model makers Bill George, Sean Casey, Larry Tan, Jeff Mann, Steve Sanders, Brian Chin, Bob Diepenbrock, and Mike Fulmer, with Martin Brenneis responsible for model electronics. Visual Effects Supervisor Kenneth Ralston remembered as follows, "The Reliant however, had to be built from scratch. Once a design was settled on, the model shop geared up and constructed it in record time. It's a gorgeous model, perfectly constructed for shooting. It's light (vacuum-formed sections over a very light aluminum interior frame), highly detailed, non-reflective (easy to shoot using the blue screen matting process), and could be easily mounted from all sides for any possible setup." (American Cinematographer, October 1982, p. 1037) Expressing his contentment of the model over the Enterprise model, Ralston gleefully added, "The Reliant is a nice squat contraption that looks a lot more believable to me. The ship takes the best of the Enterprise, rearranges it, and adds a few goodies of its own. It's much easier to have it sit there and look right. And the model is great. It's made of vacu-formed plastic and two guys can mount it on the pipes for a shot." Apart from this, Gawley's team made sure that the wiring for the internal lighting was much simpler than the one on the Enterprise model, with which the VFX team had so much trouble. (Cinefantastique, issue 44, Vol 12 #5/6, p. 55)
In constructing the model, ILM took much consideration in creating a model that could be light enough to be easily maneuvered during photography. This approach was taken because they did not want to create another bulky model, like the Enterprise built for The Motion Picture. As a result, the Reliant model was made at scale three-quarters that of the Enterprise. The model's simpler interior wiring was limited only to what was required of it as written in the script. The interior contained a metal armature that allowed it to be mounted in various positions as required for filming. The roll bar was a removable piece. A heavily modified duplicate was constructed to showcase the battle damage for later scenes in the movie.
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Gawley's team needed about four months in 1981 to build the studio model as well as all the derivative (partial) models of the class. "We started in September and finished in late December. I've been dealing with the producer, Robert Sallin, since late August. He's been very involved.", according to Gawley. (Cinefantastique, issue 44, Vol 12 #5/6, p. 55)
The high quality large scale studio model continued to be reused for over fifteen years by the producers of TNG and DS9 to represent a variety of different starships and classes, following modifications used in creating the USS Saratoga NCC-1887, USS Lantree, the USS Brattain (on the model misspelled as Brittain), the USS Bozeman, and the USS Saratoga NCC-31911. In a privately shot behind-the-scenes video, shot by Effects Supervisor Gary Hutzel at Image G (where all the footage of the model for its televised appearances were shot), it was disclosed that when the model was prepared to make its appearance as the Lantree, the effects crew were unable to make the internal lighting of the roll bar work on time, so the decision was made to have the Lantree appear without it. wbm
The model was heavily modified to represent the Soyuz-class USS Bozeman in TNG: "Cause and Effect". According to the Star Trek: The Next Generation Companion (3rd. ed., p. 195), the intention was "to make the ship a 1960s-era, pre-movie Constitution-class starship, but budget demands put an end to that when the costs of creating and filming ship, props, and costumes were added up." Simply put, what was meant was that most of the visual effects budget for the episode had already gone into the construction of the various models deemed necessary for the episode, such as the break-away model of the Galaxy-class studio model, the large nacelle model, and the shuttlebay model. Instead, the existing Miranda-class model was utilized, embellished with several add-ons to give it a differing appearance. The embellishment parts were constructed at Greg Jein's model shop. Michael Okuda has put it as follows, "It was originally hoped that a new design could be developed for the Soyuz-class U.S.S. Bozeman, but practical considerations dictated the reworking of the existing Miranda-class U.S.S. Reliant model originally built for Star Trek II. The modifications were designed by Greg Jein and Mike Okuda. The class was named for the Russian spacecraft that shuttled cosmonauts up to the Salyut and Mir space stations." (Star Trek Encyclopedia, 3rd ed., p. 457) Okuda made use of the fan-produced Ships of the Star Fleet, Volume One (pp. 84-85) to make annotations on one of the blueprints for the adjustments, eventually seen on screen.
Apart from the different add-ons to the original studio model for it to become the Soyuz-class, Gregory Jein, Inc. also constructed several debris parts as embellishments for the collision scene where the Bozeman collides with the Enterprise-D and their subsequent destruction in the episode. One of these, an, out of resin constructed aft warp nacelle fragment (presumably from the port engine, as it was this one which hit the Enterprise-D in "Cause and Effect"), measuring 16 inches, turned up as Lot 333 in the Profiles in History's The Ultimate Sci-Fi Auction of 26 April 2003, estimated at $600-$800, actually selling for the low estimate.
The add-ons for "Cause and Effects" were not permanently affixed, which was just as well as the model was again endowed with different add-ons for its appearance as the USS Saratoga (NCC-31911) in Star Trek: Deep Space Nine's pilot episode "Emissary".
After "Emissary", the model was put back together in its original configuration and returned to its birthplace, ILM, for a last time use as an unnamed ship in Star Trek Generations, without having its Brattain markings on the roll bar nor its Saratoga-markings on the saucer changed, since that was deemed unnecessary as the markings were illegible on-screen.
In this last configuration the model was, on 31 July 2002, shortly on its only public display appearance in the lobby of the Paramount Pictures Theater on the occasion of the theatrical screening of the Director's Edition of Star Trek II: The Wrath of Khan to celebrate the release of the Director's Edition DVD one week later.   The model, measuring 63×37 inches, and labeled as both the USS Saratoga (on the main body) and USS Brattain (on the roll bar), was eventually sold as Lot 985 on 7 October 2006 in Christie's 40 Years of Star Trek: The Collection auction, described as "USS RELIANT MODEL". Estimated at US$7,000-$9,000, it sold for US$50,000 ($60,000 with premium). The model was added to the collection of ScienceFictionArchives.com, an European organization that is dedicated to preserve science fiction production assets for public display purposes, such as in museums. The winner subsequently sought out Ed Miarecki's help in restoring the model to a pristine condition. 
Partial studio models Edit
Enlarged section models were constructed for close-up shots, notably the roll bar. Explosive charges and pre-packaged pieces of plastic scrap, called "nernies" (an internally-used ILM term that was also used to describe miscellaneous nondescript bits and pieces attached onto their studio model builds), that flew outwards upon detonation, were packed into the casts to give the illusion of an exploding ship. The explosion model was formed from wax to allow the pieces to be easily re-molded for additional shots and was constructed in such a way that most of the model remained intact after detonation of the charges. (Cinefantastique, issue 44, Vol 12 #5/6, p. 57) Other pieces that were constructed in a similar fashion were a port nacelle and a section of the aft deck were the impulse deflector crystal was located (these pieces are featured in the "Inside the Star Trek Archives" special on the TNG Season 2 DVD). Gawley commented, "It was very interesting and quite a challenge, to construct models so that Ken Ralston and our pyrotechnics expert, Thaine Morris, could blow them to smithereens on film, without actually damaging the [studio] model." (Cinefantastique, issue 44, Vol 12 #5/6, p. 55), to which Ralston has added, "One of the engine pods blows up. We couldn't blow up the whole pod for some reason, so I built a shape similar to it and it was more like glass blowing out of the warp nacelle. We shot that as a separate element an then printed that on top of the actual model of the Reliant, with other pieces blowing off of it. Then, when the whole nacelle blows off, that was just a bunch of explosions and a separate arm that we shot using motion control." (Star Trek: The Magazine Volume 3, Issue 5, pp. 25-26) The nacelle soldiered on in providing services as debris of the exploding USS Odyssey in DS9: "The Jem'Hadar" as well as debris of the crashed USS Olympia in DS9: "The Sound of Her Voice".
Other filming models Edit
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Apart from the high-quality large scale model, the same ILM team also built a smaller battle-damaged version for forced perspective shots in the movie (the second encounter with the Enterprise in the Mutara Nebula). It was constructed out of an AMT/Ertl Enterprise model kit, No.S970, and custom made vacuum-formed pieces, and like its bigger sister it could also be outfitted with it a battle-damaged roll bar, apart from a pristine roll bar. It was the only time that this model was used for filming purposes. Prior to being sold off at auction, it made its only tour appearance at Christie's, London on 3 August-8 August, 2006.
This model, sporting the battle-damaged roll bar, was eventually sold as Lot 984 on 7 October 2006 in the aforementioned Christie's auction. It was described as "SMALL BATTLE-DAMAGED U.S.S. RELIANT MINIATURE MODEL", estimated at US$6,000-$8,000. Measuring 17×10×4 inches, the model came with the above mentioned studio blueprint depicting the early concept for Reliant in its original "upside down" configuration. It sold for US$24,000 ($28,800 with premium). The winner, a man from London, is featured in The History Channel's documentary Star Trek: Beyond the Final Frontier.
Three so-called break-away models of the class, cast from the molds taken off the studio model, representing the USS Saratoga, were built for the DS9 premiere "Emissary". (Movie Magic, - "Models and Miniatures: A Model of Perfection") The Saratoga destruction sequence, along with the other Wolf 359 visual effects sequences, was one of the first scenes filmed for "Emissary", during pre-production on the pilot episode. The sequence was shot on a Paramount sound stage, rather than at Image G due to safety reasons over the use of explosives. Three takes were required to achieve the desired detonation - using all three breakaway models built for the sequence. (Movie Magic - "Space Effects: The Space Race")
Another physical model was constructed representing the USS Trial appearing in "The Way of the Warrior". This was a modified AMT/Ertl model kit (No.8766) of the USS Reliant. The origin of the name is a result of the rearrangement of the letters R-E-L-I-A-N-T to get T-R-I-A-L (minus the E-N). The registry was also conceived the same way, rearranging 1-8-6-4 to get 1-9-4-8. 
Visual Effects Supervisor David Stipes has made use of additional model kits, labeled "NCC-4186" and "NCC-9481",  for the closing fleet shots in "Call to Arms", footage of which, shot for that episode, was also used for consecutive episodes as stock footage, most notably in the scenes where the ships circled Starbase 375. " Yes, if Miranda = Reliants. (All model kits of Reliant) Not to confuse [the] issue, there may be another Reliant in the distance. We shot several groupings of Reliants. Some were dropped because of technical problems and some ships were removed to make the shot less confusing.", Stipes clarified. 
Aside from the filming models, camera test models of the Miranda-class, typically made out of Styrofoam, were used throughout the entire run of Star Trek, where the class was applicable. One of these, endowed with the registry "NCC 1810" was offered up for sale at a February/March 2009 It's A Wrap! sale and auction as Lot 10353, but went unsold.
Other Miranda-class derivative starship filming modelsEdit
An additional AMT model kit of the USS Reliant was utilized by Adam Buckner to "kitbash" the USS Bradford in a configuration that was reminiscent of Jennings' et al. original design intention, but with two additional under-slung cargo containers, akin to that of Franz Joseph's design for the Ptolemy-class design. Buckner stated, "(...)it's a cargo variant for carrying substantial loads. I always Imagined the rest of the main hull being converted to traditional cargo bays as well. Large doors and transporter arrays for loading cargo."  The model was constructed in order to beef out the opening scene of a retreating flotilla of Starfleet vessels in the Deep Space Nine episode, "A Time to Stand". In private correspondence with a poster on the Flare Sci-Fi Forums, Buckner has stated that he used parts from helicopter and tank model kits for most of the add-on parts (i.e. the long tubes over the shuttlebays are missiles from the helicopter kit, and the "ring" around the Saturn V cargo containers are tank treads). The top part of the Reliant weapons pod was on top of the hull. There was an Excelsior shuttlebay underneath the ship. Also add were parts that were meant to look like patches that were bolted on to cover hull damage, implying that it’s a pretty old ship that’s been haphazardly repaired. The whole idea Buckner was trying to convey with this ship was that instead of the sleek, streamlined Starfleet vessels normally featured, this was more like a tugboat that did all the grunt work.  It, however, was not present in the scene, as Buckner related, "I also have a Reliant converted Tug we built that I think dodged most photography sessions as it sat on Peter Lauritson's desk for a little too long." wbm It was however, prepared for shooting as Buckner later confirmed, "Those photos show the ship while still under construction and the top of the dish had not yet been secured while we were waiting for the neon to be fitted. It was a pain in the patoot, and after that most of that round of kit bashing was done with UV paint and UV tape for windows and nacelles." wbm
Another Miranda-class derivative model, Buckner built, and which did make it onto screen, was that of the USS Antares (NCC-9844). In the above mentioned private correspondence, Buckner further explained that it was a standard Reliant model kit with an AWACS pod made from the stand of an AMT Excelsior kit (No. 6630), and the nacelle coverings from the same kit used as torpedo launch tubes. It was actually filmed, unlike the Bradford, as a deep background ship element for use in scenes with Starbase 375 before the franchise went fully CGI. Because the scenes were just that, deep background ships circling the starbase, most of them being just standard Reliant and Excelsior kits, they did not require them to be kitbashed or battle-damaged like the ships in the opening of "A Time to Stand", and therefore needed only minimal detailing. For example, it did not have the elaborate lighting wiring that the Bradford was requested to have, the lighting as seen added in post-production, nor did it require a detailed paint job. This model was furthermore supposed to be a standard Miranda, enticing Buckner to decide of having some fun with it by adding the stand in lieu of the standard roll bar. The model, however, was filmed upside-down, so the alterations to the design were indiscernible on-screen, save for one establishing shot.  Like his other kitbash models, Buckner has retained ownership over this model as well.
Evidence of yet another Miranda-class derivative kitbashed filming model turned up in 2013, when former Visual Effects Supervisor Dan Curry provided photographs to the aforementioned Flare Sci-Fi Forums blog. His pictures showed a ship that was canonically established as the, otherwise unseen, USS Trieste (NCC-37124), though the model itself was shy of one registry digit, as it read "NCC-3724". A curious design, the model consisted of the main hull of the Reliant AMT model kit, with two Galaxy-class model kit warp nacelles attached to the dorsal side, while two ball turrets, presumably gun turrets (reminiscent of the large planet Hoth ion gun turret seen in the second Star Wars installment, The Empire Strikes Back), were placed on the locations of the bridge and impulse engine deflector crystal. Likely built during one of Deep Space Nine's kitbash rounds, by an as of yet undisclosed production staffer, the model was actually in a more advanced state of preparation for filming than Buckner's Antares was, as its paint job was more detailed, and as there was already reflective UV paint applied. Still, that model ended up unused, as there is no visual evidence of it ever appearing in an episode.
Another modified AMT model kit was seen throughout the entire run of Star Trek: Deep Space Nine, representing Benjamin Sisko's USS Saratoga. First appearing in the DS9: "The Nagus", where the model was displayed in Jake Sisko's classroom, it subsequently showed up as a display model in Benjamin Sisko's Deep Space 9 office for most of the remainder of the series.
Another commercial product was used as one of the four models mounted in front of the sail-shaped top half facade of Starfleet Headquarters on the Presidio appearing in DS9: "Homefront", "Paradise Lost". The top half of the facade was itself a maquette and the starship models were Galoob Micro Machines, most likely taken from the pewter painted "Star Trek Television Series I Box Set" and "Star Trek the Movies Collectors Edition" sets. (Star Trek: The Magazine Volume 1, Issue 13, p. 112) A Miranda-class vessels was mounted second from the right.
Yet another AMT model was slated to make an appearance, along with other models of varying starship classes, as one of the golden models in the display cases in the observation lounge of the USS Enterprise-E in Star Trek Nemesis. John Eaves built the model, filled it with resin, had it gold-plated, but failed to deliver the model in time for shooting and it was subsequently never used, as he recalled, "For the films, I was commissioned to build these models and have them gold plated at Artcraft Plating in Burbank, to be featured in the set case. Over the span of the three films, First Contact, Insurrection, and Nemesis I had to build a lot of models. For Nemesis the list grew with the request to build and plate some additional models. The Voyager, the Grissom, the Excelsior, and the Reliant. We had two cases to fill and these models were to accompany the Enterprise models. The set was constructed in an arch with a display case on either side of a large video playback display. The Enterprise models were to be on the right side and the new non Enterprise models were to be on the left. While the new ships were being plated the shooting schedule changed and the set had to be used sooner than later, because of this a double set of the Enterprise models were set in both cases. The new models barely made it back in time, but due to the amount of work involved to change them out it wasn't worth the effort, so in the end the new ships were ditched for the two sets of the Enterprises."  Eaves retained the models, "I got the other ships done in about a week and took them to the stage and he [John Dwyer] had already filled the cases twice with doubles of the original 6. It was too much work to take out and replace the dups with the new so they didn't make it on the big screen, but they did wind up at my house again.".  Eaves, citing "tough times" as reasons for reluctantly doing so, eventually sold off his gold-plated Reliant model at eBay on 13 July 2013 for US$1,000. 
Another unused display model, this time with a four-engine configuration, turned up as Lot 191 at a It's A Wrap! sale and auction. Described as "BRONZED STARSHIP MODEL", this Miranda-class derived AMT model had four nacelles, with the standard nacelles under the ship along with two others attached to the roll bar above the other nacelles, and endowed with an overall bronze hue. It sold on 19 January 2007 for US$232,50.
CGI model Edit
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Even as the original models were worn out, a new CGI model was created, making its first appearance as a deep background model in Star Trek: First Contact. A low resolution CGI model, built and rendered at ILM by Larry Tan and Paul Theren, havong used Electric Image software for animation, and Form-Z software for the model itself. wbm.
According to Visual Effects Coordinator David Stipes, the Miranda-class ships appearing in the later seasons of Deep Space Nine and Star Trek: Voyager were modeled, surfaced, and animated at Digital Muse,  debuting in DS9: "Favor the Bold". Since ILM was requested by Paramount Pictures to hand over their CGI models of the Norway-class, Saber-class, Akira-class, and Steamrunner-class, also featured in First Contact, for remodeling and remapping in the LightWave 3D software for the upcoming episode "Sacrifice of Angels", it stands to reason that Stipes referred to the CGI model of the Miranda-class ILM built, something Muse's Supervisor Bruce Branit later confirmed:
"It was the first time that anyone had actually assembled the entire Starfleet fleet in CG. Normally there were always a few ships they used for CG, and they pulled models out, and did motion control. Due to the nature of the show, there was no way they could do it with motion control. There was not enough time and not enough money. They were talking about having fifty to a hundred Starfleet vessels on screen at one time, and there was no way to pull that off in traditional ways. So we were a collecting point for anything that had been done in CG before. We brought the digital models in and converted them to LightWave 3D, which is our rendering package of choice. The Enterprise-D had been done before, but in something else, so we were able to bring the geometry in, and bring some of the maps in, but we had to rebuild it. We had all the ingredients, so we could put it together much more quickly than building it from scratch. So now we have folders with the entire fleet all lined up in the same form, so we can just load a Reliant, we can load a Defiant, we can load an Excelsior, whenever we need it. That was the first real challenge, to get all that stuff in order, and to fill the garage with useable ships." (Cinefantastique, Vol. 30, No. 9/10, p. 64)But like the others it was built at a fairly low resolution and had to be partially rebuilt and remapped.
In 2001, Robert Bonchune and Adam Lebowitz refined the model for representation in their book Star Trek: Starship Spotter. The CGI model went on to make several subsequent appearances in and onto licensed Star Trek print publications, most notably in the Star Trek: Ships of the Line calendars and their book derivative.
For the 2008 remastering of TOS: "The Ultimate Computer", the production crew of David Rossi, Denise, and Mike Okuda briefly considered using alternate ship types to represent the fleet that engaged the M-5 computer. According to Mike Okuda, "one of the ideas we considered was to have a Reliant-type configuration for a couple of the guest ships. The problem with a smaller ship is that you don't want the sense that the Enterprise is more powerful, because that would hurt the drama of the story. Ultimately, keeping all four ships the same emphasizes that the only variable is that the Enterprise is piloted by the M-5 computer." Another factor in maintaining the integrity of the original designs, according to Denise Okuda, was that "we all decided that it was best to honor the expectations of fans who had grown up seeing four Constitution-class ships."