Here is the full list of all 17 Ape Escape trophies - there are 2 bronze, 4 silver, 10 gold and 1 platinum.
Obtain the Water Net.
Collect your First Specter Coin.
Capture your First Monkey.
Collect all other Ape Escape Trophies.
Capture 50 Monkeys.
Capture 100 Monkeys.
Collect 20 Specter Coins.
Collect 40 Specter Coins.
Obtain the Monkey Radar.
Obtain the Slingback Shooter.
Obtain the Super Hoop.
Obtain the Sky Flyer.
Obtain the RC Car.
Obtain the Magic Punch.
Play the Ski Kidz Racing mini game.
Play the Specter Boxing mini game.
Play the Galaxy Monkey mini game.