Here is the full list of all 36 Cannon Brawl trophies - there are 19 bronze, 12 silver, 4 gold and 1 platinum.
Build 500 buildings.
Rank up to Captain.
Destroy 500 buildings.
Rank up to Commander.
Drop the enemy HQ in the water.
Defeat the enemy robot boss in Campaign mode.
Build 250 buildings.
Destroy 250 buildings.
Defeat the enemy airship boss in Campaign mode.
Reflect your laser off an enemy shield to damage another enemy building.
Rank up to Ensign.
Rank up to Lieutenant.
Defeat the enemy warship boss in Campaign mode.
Damage Shield Towers 5 times by shooting around their shields.
Upgrade buildings 10 times.
All Trophies Collected.
Defeat Insane AI in Battle vs. AI mode without using shields.
Win against the Insane AI by selecting the warhead as your only attack building (Banks count as an attack build).
Reflect your laser off two enemy shields to damage another enemy building.
Rank up to Vice Admiral.
Rank up to Admiral.
Rank up to Grand Admiral.
Complete the campaign on nightmare mode. You did it!
Defeat the enemy warship on nightmare mode.
Defeat the enemy airship on nightmare mode.
Destroy 4 buildings during one blasting of the Flame Thrower.
Earn 57 medals.
Destroy 1000 buildings.
Destroy 5000 buildings
Build 1000 buildings.
Build 5000 buildings.
Win a ranked game.
Win a total of 20 ranked games.
Win a total of 100 ranked games.
Win 5 ranked games in a row.
Reach 1500 rating in ranked games.