Here is the full list of all 51 Teenage Mutant Ninja Turtles: Mutants in Manhattan trophies - there are 38 bronze, 10 silver, 2 gold and 1 platinum.
Unlock all Trophies
Climb to the highest point in Karai's stage and do a little dance
Defeat 5 enemies with both Turbo Mode and Berserk activated
Dismantle Charms 30 times
Complete a stage without being spotted once (outside of Missions)
Traverse 1 mile in the Residential Area without touching the ground
Transport gold bars 30 times
Transport 30 Krang Bombs to a portal
Protect 30 ATM machines
Defeat 10 Stone Warriors with turrets in Wingnut's stage
Obtain all Collectibles
Transport 10 Chemical Weapons to a Portal
Transport 30 bags of money
Use any character at least 50 times
Upgrade 30 Charm Effects
Defeat the 4 Krangdroids with your conventional weapons in Mega Krang's stage
Defeat Rocksteady and Bebop by letting them get run over by a train
Survive the elevator section at the TCRI building without taking any damage
Eat 100 pizzas
Have Mikey do his co-op move finishing pose during the snapshots when a boss is defeated
Complete a gold bar mission without the carrier being hit even once
Obtain 50% of all Collectibles
Protect 10 Pizza Trucks
Hang out outside of an enemy hideout for at least 10 seconds
Have everyone perform a Communication Action during the snapshots when a boss is defeated
Traverse a total of 150 miles
Use Turtle Switching to defeat 1 enemy with all 4 Turtles
Disarm 30 time bombs
Detonate 100 drum cans (including proximity mines)
Obtain 25% of all Collectibles
Eat a pizza during the snapshots when a boss is defeated
Destroy a rampaging car with a drum can
Defeat 5 enemies with 1 drum can
Complete Hard mode
Defeat a Secret Boss
Enter the Pizza Room during the snapshots when a boss is defeated
Defeat 5 enemies at once by blowing up a Foot Soldier carrying a bomb bag
Play online at least once
Activate the ceiling switch during the battle with Slash
Use a Combo Attack to defeat 6 enemies at once
Complete Easy or Normal mode
Experience all of the Krang UFOs' status effects
Find the secret room in the TCRI building
Destroy 30 cameras and laser devices
Traverse a total of 32 miles
Destroy a helicopter
Have all Turtles turn into zombies simultaneously by touching contaminated water
Collect 10,000 Training Points
Equip at least one Emblem to your profile
Grind rails for 300 feet
Complete the Tutorial