Here is the full list of all 24 Ryan's Rescue Squad trophies - there are 10 bronze, 6 silver, 7 gold and 1 platinum.
Get all the trophies
Start Toy's World
Start Pirate's World
Start Prehistoric World
Start Space World
Complete a level with 2 Players
Complete the Toy's World Minigame
Complete the Pirate's World Minigame
Complete the Prehistoric World Minigame
Complete the Space World Minigame
Replay a level
Complete Toy's World
Complete Pirate's World
Complete Prehistoric World
Complete Space World
Unlock Gus
Unlock Alpha Lexa
Complete the game
Complete a level without being hit
Collect a total of 10.000 Sun Coins
Free all your friends
Get all the Surprise Eggs
Buy all the skins
Get all the Gold Coins