Here is the full list of all 47 Skylanders Trap Team trophies - there are 39 bronze, 4 silver, 3 gold and 1 platinum.
Chapter 12: Destroy 5 Da Pinchy statues
Chapter 5: Destroy 8 Troll Transports during the flying sequence
Chapter 3: Destroy 4 stone Chompy heads
Complete a captured Villain's quest
Complete Chapter 17
Chapter 17: Complete all the tile floor puzzles without falling
Complete Chapter 16
Chapter 16: Destroy 10 cacti
Complete Chapter 15
Chapter 15: Shoot down 30 Air Pirates without using rockets
Complete Chapter 14
Complete Chapter 13
Complete Chapter 12
Complete Chapter 11
Chapter 11: Ride the train to the end of the line
Complete Chapter 10
Complete Chapter 9
Collect 20 Skystones
Chapter 9: Shoot 20 Evilikin Runner during the flying sequence
Complete Chapter 8
Complete Chapter 7
Complete Chapter 6
Chapter 6: Destroy 4 stacks of pipes using the crane on Dredger’s Yacht
Complete Chapter 5
Complete Chapter 4
Complete Chapter 3
Achieve Portal Master Rank 5
Complete Chapter 2
Defeat 100 enemies in Kaos Mode
Complete Chapter 1
Unlock your first Elemental area
Earn all other Trophies
Complete Chapter 18
Chapter 4: Destroy 1 Dropship
Chapter 7: Collect 20 coins while following Marsha through the mist
Chapter 8: Knock 12 Golden Balls off the waterfall in the Meditative Pool area
Chapter 10: Travel to Splash Station without taking any damage from goo
Chapter 13: Take down every shield unit during the flying sequence
Chapter 14: Complete the arena battle without getting hit by rocket exhaust
Chapter 18: Collect 9 coins while falling down the Machine Heart
Complete Story Mode on any difficulty setting
Complete Story Mode on the Nightmare difficulty setting
Earn 50 Stars in Kaos Mode
Unlock Brock's special arena
Complete all Arena levels
Capture 10 Villains
Level up any Skylander to level 20