AFTER PLAYTHROUGH 2

Pistols:

Teapot - Quest reward from Tiny Tina "You are cordially invited"(unique)
Judge - Drops from Assassin Oney in Southpaw Steam and Power(Unique)
Rubi - Quest reward from Moxxi "Rakkaholic Anonymous" (Unique)
Fibber - Quest reward from "A Real Boy" (Unique)
Veritas - Quest reward from "Clan Wars: Wakey Wakey" (Unique)
Flint's Tinderbox - Drops from Captain Flint (Unique)
Gwen's Head - Found in a box near the Buzzard camp in The Dust (Unique)
Lady Fist - Quest reward from Una Mailbox in "Uncle Teddy" (Unique)
Dahlminator - Quest reward from "The Lost Treasure" (Unique)
Law - Quest reward from "Won't get fooled again", Drops from the Sheriff of Lynchwood (Unique)

Shotguns:

Thre Dog - Drop from Assassin Rouf in Southpaw Steam and Power (Unique)
Triquetra - Quest reward from "Clan Wars: End of the Rainbow" (Unique)
Moxxi's Heartbreaker - Quest reward from "Safe and Sound" (Unique)
Shotgun 1340 - Quest reward from Marcus during "Out of Body Experience (Unique)
Blockhead - Drops from Badass Creepers (Unique)
Octo - Quest reward from "Slap Happy" (Unique)
Roksalt - Quest reward from "Splinter Group" (Unique)
Landscaper - Quest reward from "Family Fued: Dueling Banjos and Fiddles" (Unique)
Tidal Wave - Quest reward from Hyperion Mailbox in "Uncle Teddy" (Unique)
Teeth of Terramorphous - Drops from Terramorphous

Assault Rifles:

Scorpio - Quest reward from "Bearer of Bad News" (Unique)
Evil Smasher - Quest reward from "The Chosen One" (Unique)
Hail - Quest reward from "Bandit Slaughter: Round 5" (Unique)
Stomper - Quest reward from "Hungery Like the Skag" (Unique)

Sniper Rifles:

Buffalo - Quest reward from "Demon Hunter" (Unique)
Fremington's Edge - Drops from Assassin Reeth (Unique)
Trespasser - Quest reward from "Animal Rights" (Unique)
Sloth - Quest reward from Mordecai in "Rakkoholics Anonymous" (Unique)
Morning Star - Quest reward from "Hyperion Contract #873" (Unique)
Chère-amie - Quest reward from "Hyperion Slaughter: Round 5" (Unique)

Rocket Launchers:

Roaster - Quest reward from "Note to Self-Person" (Unique)
Creamer - Quest reward from "Creature Slaughter: Round 5" (Unique)
The Hive - Saturn, Arid Nexus (Unique)

SMGs:

Bane - Quest reward from "The Bane" (Unique)
Miss Moxxi's Good/Bad Touch - Both rewards from tipping Moxxi, no set amount, can be got constantly (Unique)
Lascaux - Found in a pond in the caves in Frostburn Cavern, respawns (Unique)
Chulainn - Quest reward from "Clan Wars" (Unique)
Commerce - Drops from Assassin Wot in Southpaw Steam and Power (Unique)
Bone Shredder - Drops from Bonehead 2.0 in Arid Nexus (Unique)

Shields:

Order - Quest reward from "BFFs" (Unique)
Love Thumper - Quest reward from "Best Mother's Day Ever" (Unique)
Deadly Bloom - Quest reward from "Helping Out Overlook" (Unique)
Aequitas - Quest reward from "Clan Wars: Wakey Wakey" (Unique)
1340 Shield - Quest reward from Dr. Zed in "Out of Body Experience" (Unique)
Pot o' Gold - Quest Reward from "Clan Wars: End of the Rainbow" (Unique)
Flame of the Firehawk - Quest reward from Lilith's Cult Mission (Unique)

Grenades:

Kiss of Death - Quest reward from "Hell Hath No Fury" (Unique)
Fuster Cluck - Quest reward from "The Pretty Good Train Robbery" (Unique)
Sky Rocket - Unlocked with Special Editions of BL2
Breath of Terramorphous - Drops from Terramorphous

Relics:

Sheriff's Badge - Drops from Sheriff of Lynchwood (Unique)
Deputy's Badge - Quest reward from "Showdown" (Unique)