The shotgun is the primary weapon used by Sarge. He begins to carry it around at the start of season 2. Sarge uses it because he claims long range weapons are unmanly. He has however been seen to use an assault rifle from time to time while fighting Blues. He killed several Wyoming clones with it at the end of season 5.
Grif and Simmons also occasionally, but very rarely, use a shotgun. Simmons used one in season 1 to guard Tex after her capture, and after Sarge "died" in season 5 so he could be the leader of the Reds. Grif also used Sarge's shotgun in an attempt to revive him after he was shot in the head, by hitting him in the chest with it.
In Recreation, Sarge and Grif were trapped in the Sandtrap minefield with Grif slightly up hill from Sarge. Grif mocks Sarge into shooting at him, knowing that the gun can't reach. Sarge only gets angrier.
In the Zombie PSA, Sarge says he carries a shotgun at all times in case of a zombie attack. This is shown when Sarge brings his shotgun with him during his vaction in Holiday Plans, Part Three: Chestnuts Roasting...
In the flashback of Tex's assault on Sidewinder, Church had a shotgun and tried to shoot Tex multiple times. In Revelation Episode 10, Tex takes Sarge's shotgun from him and attempts to shoot Grif in the face, only to find that there was no ammo left. She then swings the shotgun right into Grif's balls, causing Grif more pain and sending him flying into Simmons. The Shotgun is also briefly seen in the Season 9 trailer, where York has one slung on his back when he, North Dakota, and Tex are shown to be surrounded by enemy forces.