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
Play a level in co-op
Use all the Minikit vehicles
Get 100% (Single Player Only)
Unlock all characters (Single Player Only)
Unlock all female heroes and villains. (Single Player Only)
Collect all the gold bricks (Single Player Only)
Climb to the top of Wayne tower while riding a Gorilla and playing as a female character.
Use the Gotham City Metro
Complete the Bonus level
Unlock all Justice League characters (Single Player Only)
Rescue all Citizens in Peril (Single Player Only)
Defeat Sinestro as Green Lantern
With Brainiac, defeat any LexBot
Get Super Hero in all levels (Single Player Only)
Obtain more than 10,100,000,000 Studs (Single Player Only)
Get 50% (Single Player Only)
Unlock all the Bosses (Single Player Only)
Defeat Zod as Superman
Collect all the red bricks (Single Player Only)
Test a custom character
Complete story level 15
Complete story level 14
Complete story level 13
Complete story level 12
Complete story level 11
Complete story level 10
Complete story level 9
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