There are 12 Trophies that can be collected.

Bronze

Shoot Some Robots
Kill 100 robots!

Greetings, Earth Citizens
Beat a mission wearing the astronaut helmet and backpack

Lookie, Many Stars
Get 75 stars!

Dominate at Fatherhood
Beat a mission wearing the baby carrier!

Slotally Tober Saughter
Complete a mission on hard or insane without dying or drinking juice!

Punch Many Bullets
Punch two bullets, killing two robots, with one melee swing!

Get Over It
Revive other players 8 times!

Make Many Friends
Have someone you invite join you in an online game!

Number One Tugnut
Come in 1st in a four player mission!

Silver

Happy 50th, Walter
Reach level 50!

Lower Carbon Footprints
Destroy a robot overlord in multiplayer!

Gold

Shoot Many Robots
Kill 10000 robots!

http://www.ps3trophies.com/forums/playstation-trophy-news/13235-shoot-many-robots-trophy-list.html