Here is the full list of all 10 Space Pirate Trainer trophies - there are 3 bronze, 6 silver and 1 gold.
Reach Wave 10
Win the game
Zap 25 Droids with the Tesla Coil in one game
Destroy your first Heavy Droid
Destroy the Ray Boss
Pull a Droid into its own Laser
Reach Wave 20 without using a Shield
Get a Multiplier Combo of 12
Survive Wave 23 for one minute without using your Gun, Shield or Volton
Destroy the Mother Droid