Here is the full list of all 39 Nex Machina trophies - there are 23 bronze, 11 silver, 4 gold and 1 platinum.
Complete Arcade mode on Experienced difficulty
Complete Space Station in arcade mode on any difficulty
Destroy every Secret Beacon in a single world
Beat Machine City on Experienced difficulty in under 5:45
Dash through 3 beams with single dash
Reach a multiplier of 50
Reach a human combo of 50
Clear a single world on Experienced difficulty or higher without getting hit
Clear a single world while succeeding in every level end dash on any difficulty
Defeat the Beamtron on Experienced difficulty with one or more enemies still alive
Complete a single world on Experienced difficulty without saving any humans
Earn 3 bronze medals in the Arena
Earn 6 silver medals in the Arena
Earn 9 gold medals in the Arena
Complete Future Labs in arcade mode on any difficulty
Complete Arcade mode on Rookie difficulty
Complete Machine City in arcade mode on any difficulty
Deal a killing blow to a boss with a Sword attack
Complete Fire Caverns in arcade mode on any difficulty
Complete Crystal Mountain in arcade mode on any difficulty
Kill every Disruptor in a single world
Succeed in 10 level end dashes in a row on any difficulty
Finish 5 levels in a row without movement, dashing allowed
Discover a secret level
Complete Techno Forest in arcade mode on any difficulty
Complete a feat
Fully power up the Hero
Reach a human combo of 5
Save a human while it's being harvested
Kill a Disruptor
Complete all trophies
Complete Arcade mode on Veteran difficulty
Complete Techno Forest on Master difficulty
Complete Arcade mode on Experienced difficulty without continuing
Finish a level without firing your primary or your secondary weapon
Save all Secret Humans in a single world on Experienced difficulty or higher
Save all Humans and Secret Humans in a single world on Veteran difficulty
Complete 50 feats
Complete 100 feats