Successfully complete one of the following tasks to get a trophy:

Tanks Very Much (Silver): Achieve level 10 as and gain the Tank Role.
Mission Control (Silver): Achieve level 10 as and gain the Controller Role.
The Healing Touch (Silver): Achieve level 10 as and gain the Healer Role.
The Queen's Favorite (Gold): Achieve level 30 as a villain mentored by Circe.
Criminal Mastermind (Gold): Achieve level 30 as a hero mentored by Lex Luthor.
The Joker in the Deck (Gold): Achieve level 30 as a villain mentored by the Joker.
Knight for Justice (Gold): Achieve level 30 as a hero mentored by Batman.
Warrior of Truth (Gold): Achieve level 30 as a hero mentored by Wonder Woman.
Champion of Earth (Gold): Achieve level 30 as a hero mentored by Superman.
Acrobatic Ace (Gold): Achieve level 30 as an acrobat.
Fearless Flier (Gold): Achieve level 30 as a flier.
Speedster Supreme (Gold): Achieve level 30 as a speedster.
Platinum Pace (Gold): Earn a platinum medal in any race.
Platinum Trophy (Platinum): Earn every other trophy in the game.

The following trophies require the "Fight For The Light" bonus downloadable content:

Parole Denied (Bronze): Defeat Red Lantern Vice, Yellow Lantern Lyssa Drak and Evil Star in the Oan Sciencells.
Parole Granted (Bronze): Defeat Red Lantern Vice, Green Lantern Guy Gardner and Evil Star in the Oan Sciencells.
Not that Repulsive (Bronze): Defeat Vice in the Coast City Duo without restarting the repulsor.
Willpower Outage (Bronze): Defeat the Scion of Will before the Sliceclaws power it up.
Illogical Reasoning (Bronze): Beat the Final Confrontation as a villain without using any Logic Batteries within the Hangar at Ferris Aircraft in Coast City.
Low Battery (Bronze): Beat the Final Confrontation as a hero without using any Logic Batteries within the Hangar at Ferris Aircraft in Coast City.
Conviction (Bronze): Complete Oan Sciencells without being knocked out.
Intergalactic Planetary (Silver): Collect all styles in the Sector Agent set.
Planetary Intergalactic (Silver): Collect all styles in the Sector Incendiary set.
Fight for the Light (Silver): Complete Coast City Duo, STAR Labs Alert, Coast City Alert and Oan Sciencells Alert.

The following trophies require the "Lightning Strikes" bonus downloadable content:

Rogues' Gallery (Silver): Complete all of the Rogues' Wanted Posters.
Crash of the Titans (Silver): Complete all of the Titans' Bounties.
Two Tickets to Paradox (Silver): Seal both types of Massive Speed Force Ruptures.
Prime Timeline (Bronze): Complete the Massive Speed Force Rupture Event.
Reap the Rewards (Silver): Defeat 100 Paradox Reapers in Central City.
Play On (Bronze): Complete the Cosmic Treadmill: Flashback Duo without destroying any Sonic Emitters.
For My Next Trick! (Silver): Defeat Abra Kadabra while all the Riot-Bots are active without being knocked out.
Quickly, Into the Past (Bronze): Complete the Cosmic Treadmill: Flashback Duo in under 7 minutes.