Here is the full list of all 44 The Last Kids on Earth and the Staff of Doom trophies - there are 32 bronze, 8 silver, 3 gold and 1 platinum.
Collect all trophies
Revive a friend in multiplayer
Reach level 5
Reach level 10
Reach level 15
Reach level 20
Complete all main and secondary quests in District 1
Complete all main and secondary quests in District 2
Complete all main and secondary quests in District 3
Complete all main and secondary quests in District 4
Defeat the vine claw
Defeat Scrapken
Defeat Spikipede
Defeat King Wretch
Defeat the Slime Impostors
Defeat Meathook
Defeat Hairy Eyeball
Defeat Neon Bosses
Craft a tree house turret
Craft all tree house turrets
Fully upgrade all tree house turrets
Craft a weapon
Craft all weapons
Fully upgrade all weapons
Craft an armor
Craft all armors
Fully upgrade all armors
Upgrade BigMama to level 2
Upgrade BigMama to maximum level
Upgrade tree house to level 2
Upgrade tree house to maximum level
Inflict 5000 damage points with Rover
Have Skaelka perform 5000 spins
Have Bardle heal 5000 health points
Have Biggun absorb 5000 damage points
Stun 250 enemies using Jack's BooMerang
Defeat 250 enemies using June's Toxic Frisbee
Use Dirk's Sticky Grenade against 250 enemies
Distract 250 enemies using Quint's Cymbal Monkey
Complete the bestiary
Complete a Big Mama challenge
Complete all Big Mama challenges
Find a comic book cover
Find all comic book covers