There are 14 Trophies that can be collected.

BRONZE

Slaughterific
Destroy 1,000 bad guys

Close Encounters
Make a nearby bad guy envious of your drill

Tolkienesque
See everything

Home Potato
Spend 20 seconds in the heart of an asteroid

Averagescorer
Get a highscore of 50,000 on each mode

Lucky number Seven
Smash through difficulty level 7

Attack of Boomer V
Destroy 5 enemies at once using a Boomer

x100
Get a x100 multiplier!

Check out my pad
This ship is fast and powerful

Check out my shop
Buy the shop!

Black Holed
Peacefully survive a Black Hole!

SILVER

Massacrertastic
Destroy 20,000 enemies

Highscorer
Get a highscore of 125,000 on each mode

GOLD

King of the skies
Classic Mode – Defeat all bosses in one round!

http://www.ps3trophies.org/game/big-sky-infinity-vita/trophies/