Here is the full list of all 16 Cake Invaders (EU) trophies - there are 5 silver, 10 gold and 1 platinum.
Defeat an enemy for the first time.
Unlock all other trophies.
Lose your first game.
Gain a skill for the first time.
Increase the crew to two.
Increase the crew to five.
Kill 50 of them with the increased damage of the bullets.
Shoot 100 bullets while the rate of fire is increased.
Shoot 200 bullets while the rate of fire is increased.
Kill two or more enemies at once with a single laser shot.
Clear your first wave.
Reach the 3rd wave.
Reach the 6th wave.
Reach the 9th wave.
Reach the 12th wave.
Endure 3 waves with 1 piece of baumkuchen remaining.