There are 15 Trophies that can be collected.

BRONZE

Welcome to Flying Hamster!
Finish Stage 1 (Normal Mode)

Finish Normal Mode
Clear game in Normal Mode

Top score
Reach first position in the local Hi-score table

Millionaire score
Score 1.000.000 points

Screen clear using a berry
Kill hen and all crows in Stage 1

Screen clear using a berry #2
Kill all bats using a berry on the wall in Stage 2

Column breaker
Destroy all columns in Stage 2 (Normal Mode)

Kill that ram!
Kill jumping ram from stage 1 (Normal Mode)

Kill that robot!
Kill jetpack robot from stage 3 (Normal Mode)

Avoid them all!
Finish any level without firing (except for the boss) with 1 credit

Hardcore spitter
Finish any level using spit weapon only

Hamster Master
Go through Stage 5-2 (Normal Mode) without losing any HP

Adventurer
Destroy secret passage from stage 2 (Normal Mode)

SILVER

1 Credit Normal Mode
Clear game in Normal Mode using 1 credit

GOLD

1 Credit Hard Mode
Clear game in Hard Mode using 1 credit

http://www.ps3trophies.org/game/the-flying-hamster-hd-vita/trophies/