There are 36 Trophies that can be collected.

Bronze

Brave New     
Completed the Hatching process for the first time     

Twins    
Hatched two EyePets at the same time     

Looking Good    
Styled your EyePet for the first time     

Trainer    
Taught your EyePet a Trick     

Engineer    
Created a customised Digger     

Traveler   
Created a customised UFO     

Captain    
Created a customised Boat     

Mariner    
Created a customised Submarine     

Astronaut    
Created a customised Rocket     

Pilot    
Created a customised Helicopter     

Crafty    
Created a card in Paper Craft     

Customize   
Created a sticker in Paper Craft and used it in the Creativity Centre     

Building Blocks    
Customized an object in Soft Play Creation     

Spit and Polish    
Washed your EyePet for the first time     

Full to the Brim    
Fed your EyePet for the first time     

Check Up    
Performed a Health Check on your EyePet for the first time

Silver

Bopper    
Completed Frog Bop and bopped each frog type at least once     

Conductor    
Completed Musicopters without getting a sequence wrong     

Space Dash    
Completed Space Race without a hitting an asteroid     

Sky Writer    
Completed Star Tracer and achieved at least a 1000 points.     

Sea Skills    
Completed Coin Collector without being stunned by the Eel     

Ring Runner    
Completed Sub Run without missing a bubble ring     

Savior   
Completed Rescue and saved at least five of each type of alien     

Full Tank    
Completed Rocket Refuel filling up at least three different fuel slots     

Fishing King    
Completed Hook a Duck and hooked at least three of each type of rubber duck     

Green Fingers    
Completed Lily Blossom and opened at least three of each type of lily     

Swish    
Completed Digger Hoops and scored at least three baskets of each colour     

Good Aim    
Completed Ricochet Range with at least 500 points     

Dedicated Follower of Fashion    
Customised one of each type of clothing and then dressed your EyePet in them     

Pet Whisperer    
Taught the EyePet every trick in training     

Craft Box Complete    
Bought all the Sticker and Stencil packs     

Pet the Builder    
Used every Soft Play shape in a single scene

Gold

Collectormanic    
Bought every item in the game     

Contender    
Beat a friend in every challenge     

Shopaholic    
Had your EyePet wear every item of clothing in the game at least once

Platinum

Platinum Pet    
Unlock all other trophies in the game

http://www.ps3trophies.org/game/eyepet-and-friends/trophies/