Here is the full list of all 18 In Space We Brawl trophies - there are 16 bronze and 2 silver.
Complete 1000 fights
Fire 500,000 shots
Kill 2,000 players
Kill 250 players with the Quantic Slinger
Kill 4,000 aliens
Kill 50 players without using weapons
Kill 250 players with the Plasma Sword
Obtain Old School achievement 50 times
Win 25 1 vs 3 matches
Collect 5 defensive pods at the same time
Complete all the challenges with S rank
Unlock all weapons
Complete all the 21 challenges
Unlock all spaceships
Play 100 mirror matches
Commit suicide 50 times
Fight in every single sector of each location
Complete 10 matches in less than 3 seconds