Here is the full list of all 10 Quiplash trophies - there are 5 bronze, 4 silver and 1 gold.
Win all battles in Rounds 1 and 2
Write "meatloaf candy" and win a battle with it
Get 70% or more of the votes in The Last Lash
Win both battles in a round
Get a Quiplash! (sweep votes in a 5+-player game)
Play a game with an Audience
Enter the same thing as your opponent
Write a quip that Schmitty actually responds to
Successfully enter both your quips in a round
Play an 8-player game