Big Bang 50
Complete story mode on Easy difficulty
Bigger Bang 50
Complete story mode on Normal difficulty
Biggest Bang 50
Complete story mode on Hard difficulty
Party Pooper 10
Time to break up this party
Freeflow Combo 20 10
Complete a combo of 20 moves (any play mode)
Freeflow Combo 40 10
Complete a combo of 40 moves (any play mode)
Night Glider 5
Glide continuously for over 100m
Rope-A-Dope-A-Dope 10
String up one henchman and drop him to surprise a second (any play mode)
Mano-A-Mano 10
Take on a beast in hand to hand combat
Catch! 5
Catch a Batarang (any play mode)
Freeflow Combo 5 5
Complete a combo of 5 moves (any play mode)
Freeflow Combo 10 5
Complete a combo of 10 moves (any play mode)
Freeflow Perfection 10
Perform a perfect combo including all of Batman's combat moves (any play mode)
Freakshow Rodeo 10
Ride a beast and unleash its power
Freeflow Bronze 10
Achieve 8 medals on combat challenges
Freeflow Silver 25
Achieve 16 medals on combat challenges
Freeflow Gold 50
Achieve 24 medals on combat challenges
Predator Bronze 10
Achieve 8 medals on predator challenges
Predator Silver 25
Achieve 16 medals on predator challenges
Predator Gold 50
Achieve 24 medals on predator challenges
Invisible Predator 10
Complete one predator challenge by using only Silent Takedowns and without being detected
Flawless Freeflow Fighter 10
Complete one combat challenge without taking damage
Crack The E Nigma 20
Solve every riddle on the island
Arkham Analyst 20
Solve 5% of Riddler challenges
Cryptic Investigator 20
Solve 10% of Riddler challenges
Lateral Thinker 20
Solve 25% of Riddler challenges
Mystery Solver 20
Solve 40% of Riddler challenges
Conundrum Cracker 20
Solve 55% of Riddler challenges
Mental Athlete 20
Solve 70% of Riddler challenges
Riddle Resolver 20
Solve 85% of Riddler challenges
World's Greatest Detective 20
Solve Arkham's biggest mystery
Perfect Knight 75
100% Complete
[via Xbox360Achievements]

The PS3 trophies may be more cause of the Joker