Here is the full list of all 34 PlayStation Vita Pets trophies - there are 20 bronze, 7 silver, 6 gold and 1 platinum.
Collect all of the trophies
Find all of the King's coins
Sniff 'n' dig in all locations
Collect all of the King's insignia
Jump up 200 times
Solve the final castle puzzle
Find Cosmo's Golden Tunic
Find all of Cosmo's collection
Explore the whole map
Successfully travel all routes around the mine cart ride
Finish the mine cart ride wearing Cosmo's jacket
Solve the final mines puzzle
Solve the final maze puzzle
Learn all of the dog skills to level 3
Reach 10,000 Buddy Points
Solve the final caves puzzle
Complete the entire obstacle course wearing the red superhero outfit
Complete the entire obstacle course for the first time
Sniff 'n' dig up 75 items
Play all of the activities
Sniff 'n' dig up 30 items
Catch the flying disc 5 times in the air and in succession
Learn all of the tricks
Reach 5,000 Buddy Points
Walk to the castle bridge
Solve the cemetery puzzle
Tug 30 objects
Try on a matching outfit for the first time
Shower for the first time
Fetch 20 sticks
Reach 1,000 Buddy Points
Reach 500 Buddy Points
Try on a bodysuit for the first time
Try on an item of clothing for the first time