(covers information from several alternate timelines)
In 1996, two militia members in Arizona, armed with shotguns, captured Chakotay and B'Elanna Torres after their shuttlecraft crash landed nearby. The two, named Porter and Butch, then employed the shotguns while attempting to defend the prisoners and the captured ship from law enforcement. (VOY: "Future's End, Part II")
In the 2020s, San Francisco police were armed with shotguns, which they carried with them as they patrolled the Sanctuary Districts. During the Bell Riots of 2024, the residents who held the government personnel hostage used acquired police shotguns to ensure their cooperation. (DS9: "Past Tense, Part I", "Past Tense, Part II")
In 2063 in the mirror universe, Zefram Cochrane used a shotgun to kill the first Vulcan who set foot on the planet Earth. The crowd who had gathered to witness the arrival of the Vulcans to Earth also used shotguns to shoot the other occupants of the T'Plana-Hath.
In 2155, Commander Jonathan Archer, after reliving Captain Maximilian Forrest of the ISS Enterprise, showed Cochrane's shotgun, kept in the Captain's ready room, to Vulcan science officer and commander T'Pol, to illustrate the fact that her people were under Terran oppression and that T'Pol was under Archer's command. (ENT: "In a Mirror, Darkly")
A holodeck program of the Ancient West, created by Alexander Rozhenko and Reginald Barclay in 2369, featured a bandito character who was armed with a double-barreled shotgun. (TNG: "A Fistful of Datas")