Here is the full list of all 38 LEGO Batman 2: DC Super Heroes trophies - there are 25 bronze, 8 silver, 4 gold and 1 platinum.
Collected all of the Trophies
Defeat Sinestro as Green Lantern
Climb to the top of Wayne tower while riding a Gorilla and playing as a female character.
Get 100% (Single Player Only)
Unlock all female heroes and villains. (Single Player Only)
Unlock all characters (Single Player Only)
Collect all the gold bricks (Single Player Only)
Use all the Minikit vehicles
Rescue all Citizens in Peril (Single Player Only)
Obtain more than 10,100,000,000 Studs (Single Player Only)
Collect all the red bricks (Single Player Only)
Complete the Bonus level
Unlock all Justice League characters (Single Player Only)
Unlock all the Bosses (Single Player Only)
Get Super Hero in all levels (Single Player Only)
Complete story level 15
Complete story level 14
Complete story level 13
Complete story level 12
Get 50% (Single Player Only)
With Brainiac, defeat any LexBot
Defeat Zod as Superman
Complete story level 11
Complete story level 10
Complete story level 9
Use the Gotham City Metro
Test a custom character
Complete story level 8
Complete story level 7
Complete story level 6
Complete story level 5
Fly with Superman
Complete story level 4
Do a finishing move
Complete story level 3
Complete story level 2
Complete story level 1
Play a level in co-op