Here is the full list of all 32 LEGO Batman 2: DC Super Heroes trophies - there are 16 bronze, 9 silver, 6 gold and 1 platinum.
Collect all the Villain characters.
Complete every arena in Justice League Mode.
Collect all the Hero characters.
Unlock all the Ability packs for the Character Customiser.
Collect all the Red Bricks.
Finish a level without losing all your hearts.
Collect all the Sound Bite tokens.
Stun an enemy using Commissioner Gordon's trusty weapon and finish them. (Single Player Only)
Take down Batman with Bane. (Single Player Only)
As Catwoman, use the Stealth Takedown ability to attack Batman. (Single Player Only)
Stun 100 enemies.
Swap characters using Super Freeplay.
Complete The Final Battle.
Complete Wayne Industries.
Help an enemy off a ledge in Wayne Industries.
Complete Brawl at City Hall.
Complete Gotham Metro.
Complete Robot Sky Battle.
Complete Attack on LexCorp.
Complete Assault the VTOL.
Complete The Batcave.
Complete Juggernaut Chase.
Complete ACE Chemicals.
Create your first character in the Character Customiser.
Complete Arkham Asylum.
Complete Arkham Estate.
Using the Batman glide ability, spend 5 seconds or more in the air.
Achieve a True Hero stud total.
Complete The Joker Getaway.
Complete Gotham Theatre.
Collect all trophies.
Win gold in every arena in Justice League Mode.