The USS Unicorn (NCC-85117) was a Federation starship that was in service with Starfleet in the mid-24th century.
In 2364, prior to stardate 41603.7, this ship, along with the USS Sherlock Holmes, received an order from Admiral Gregory Quinn of Starfleet Operational Support Services.
At the end of that year, Data read these orders and several others to investigate claims of a conspiracy among the highest members of Starfleet Command. (TNG: "Conspiracy")
The ship was never mentioned in dialog; this name came from a mission order. The captain's name was Terry Ann something.