Here is the full list of all 21 Awesomenauts Assemble! trophies - there are 21 bronze.
Reach max level.
Destroy 25 turrets.
Win game sessions with 8 different characters.
Collect 35 Solar on a drop-pod drop.
Kill 1000 droids/creeps.
Taunt an enemy after he killed you.
Taunt an enemy after killing him.
Make the most kills of both teams in an online game session.
Win a game in less than 10 minutes.
Win a game session with 2 friends splitscreen.
Earn a total of 10000 Solar ingame (late game bonus does not count).
Kill an enemy Awesomenaut without buying any items.
Kill 25 enemy Awesomenauts.
Get killed by a creep.
Unlock 3 characters.
Get first blood.
Win an online game session.
Play a game session with friends invited through the game session setup.
Have more than 500 Solar at once (late game bonus does not count).
Kill 3 enemy Awesomenauts without dying.
Play against a team with the same Awesomenauts.