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