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 trophies.
Win gold in every arena in Justice League Mode.
Collect all the Villain characters.
Complete every arena in Justice League Mode.
Unlock all the Ability packs for the Character Customiser.
Collect all the Hero characters.
Collect all the Red Bricks.
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)
Complete The Final Battle.
Complete Wayne Industries.
Help an enemy off a ledge in Wayne Industries.
Complete Brawl at City Hall.
Complete Gotham Metro.
As Catwoman, use the Stealth Takedown ability to attack Batman. (Single Player Only)
Complete Robot Sky Battle.
Finish a level without losing all your hearts.
Swap characters using Super Freeplay.
Complete Attack on LexCorp.
Complete Assault the VTOL.
Complete The Batcave.
Complete Juggernaut Chase.
Complete ACE Chemicals.
Complete Arkham Asylum.
Stun 100 enemies.
Complete Arkham Estate.
Complete The Joker Getaway.
Create your first character in the Character Customiser.
Complete Gotham Theatre.
Achieve a True Hero stud total.
Using the Batman glide ability, spend 5 seconds or more in the air.