Here is the full list of all 33 ONRUSH trophies - there are 19 bronze, 7 silver, 6 gold and 1 platinum.
Perform 1000 'NEAR ALLY' actions
Personalise all 8 Classes
Play 10 LOCKDOWN Matches
Play 10 COUNTDOWN Matches
Superstar - Unlock 'Snowbound'
Reach player Level 30
Feature in the player spotlight 10 times
Earn 50 'HUNTER' Medals
Play 10 SWITCH Matches
Play 10 OVERDRIVE Matches
Superstar - Unlock 'Starpower'
Reach player level 15
Earn 10 Crashtags
Superstar - Unlock 'Regeneration'
Reach player level 5
Superstar - Unlock 'Uprising'
Perform a 'CRUSHED!' Takedown
Superstar - Start 'Origins'
Smash 5 Tombstones in a single match
Rush for the first time
Takedown an opponent!
Introduce yourself to the fodder
Use boost for the first time
Obtain all trophies in ONRUSH™
Perform a 'VOMIT COMET' action
Personalise all 12 Characters
Collect 250 or more Gear Items
Earn 5 Ubertags
Superstar - Unlock 'Superstars'
Superstar - Win the Founders' Trophy
Earn 50 'CHECKMATE!' Medals
Earn 100 Gold 'GAME TIME' Medals
Superstar - Max out your Superstar Rating