Complete the following tasks to unlock PlayStation 3 trophy rewards.

Trophy	How to unlock
A Fistful of Arkham (Bronze)	Complete all of the exterior missions at Arkham Island.
Ace Chemicals Ace (Bronze)	Complete all of the exterior missions at Ace Chemicals.
All Their Base (Bronze)	Win 1000 Lair PvP Battles.
Factory Direct (Bronze)	Complete all of the exterior missions at Steelworks.
Interior Designer (Bronze)	Unlock Free Placement Mode.
It's a Shutout! (Bronze)	Win 5 consecutive Lair PvP Battles without letting your opponents win a single round.
Riot Squad (Bronze)	Complete all of the exterior missions at Stryker's Island.
The World of Fashion (Gold)	Complete the feats: I Can HAZMAT, Don't Pick Up Hitchhikers, Lost and Foundry, and Maximum Security.