Here is the full list of all 24 Laser League trophies - there are 4 bronze, 12 silver, 7 gold and 1 platinum.
Reach Level 20
Reach Mastery rank 2 with all base game Classes
Win 50 matches (Online Play)
Activate 5000 Laser nodes (Online Play)
Revive 500 teammates (Online Play)
Teleslice 2 or more opponents in a single Snipe ability use (Online Play)
Hit 3 or more opponents in a single Smash ability use (Online Play)
Eliminate 2 opponents with a single expanding Laser (Online Play)
Reach Level 10
Eliminate 1000 opponents (Online Play)
Slice 2 or more opponents in a single Blade ability use (Online Play)
Finish a match as MVP 10 times (Online Play)
Traverse 4 or more opposing Lasers in a single Ghost ability use (Online Play)
Shock 3 or more opponents in a single Shock ability use (Online Play)
Steal 4 or more Laser nodes in a single Thief ability use (Online Play)
Finish a match with the highest Revive streak 5 times (Online Play)
Finish a match with the highest Elimination streak 5 times (Online Play)
Win a match (Online Play)
Level-up for the first time
Win a match without the opposing team scoring a point (Online Play)
Complete the Tutorial
Earn all other Trophies
Reach Prestige rank 01
Reach Mastery rank 5 as any Class