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
Complete Story Mode on the Nightmare difficulty setting
Chapter 14: Complete the arena battle without getting hit by rocket exhaust
Chapter 10: Travel to Splash Station without taking any damage from goo
Chapter 8: Knock 12 Golden Balls off the waterfall in the Meditative Pool area
Chapter 6: Destroy 4 stacks of pipes using the crane on Dredger’s Yacht
Complete all Arena levels
Unlock Brock's special arena
Complete Story Mode on any difficulty setting
Complete Chapter 18
Chapter 18: Collect 9 coins while falling down the Machine Heart
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
Chapter 13: Take down every shield unit during the flying sequence
Complete Chapter 12
Chapter 12: Destroy 5 Da Pinchy statues
Complete Chapter 11
Chapter 11: Ride the train to the end of the line
Complete Chapter 10
Earn 50 Stars in Kaos Mode
Level up any Skylander to level 20
Complete Chapter 9
Chapter 9: Shoot 20 Evilikin Runner during the flying sequence
Complete Chapter 8
Collect 20 Skystones
Defeat 100 enemies in Kaos Mode
Complete Chapter 7
Chapter 7: Collect 20 coins while following Marsha through the mist
Achieve Portal Master Rank 5
Complete Chapter 6
Complete Chapter 5
Chapter 5: Destroy 8 Troll Transports during the flying sequence
Complete Chapter 4
Capture 10 Villains
Chapter 4: Destroy 1 Dropship
Complete Chapter 3
Chapter 3: Destroy 4 stone Chompy heads
Complete Chapter 2
Complete a captured Villain's quest
Complete Chapter 1
Unlock your first Elemental area