Here is the full list of all 32 SUPER BOMBERMAN R trophies - there are 18 bronze, 8 silver, 5 gold and 1 platinum.
Played 1,000 Multiplayer Battles
Won a battle with each of the 5 Dastardly Bombers
Beat Expert on Story Mode
Beat Beginner on Story Mode
Beat Veteran on Story Mode
Won with a Revenge Cart in a 4 Player Multiplayer Battle
Created a Room and Battled 100 times
Beat Black Hole on Story Mode
Beat Planet Scrapheap on Story Mode
Beat Planet Lalaland on Story Mode
Beat Planet Brainwave on Story Mode
Maximized your movement speed
Played 100 Multiplayer Battles
Tied with a Time Up
Over 10 Bombs Exploded as a Combo
Won a battle with each of the 8 Bomberman Brothers
Created a Room and Battled 10 times
Beat Planet Timbertree on Story Mode
Beat Planet Technopolis on Story Mode
Carried the MAX number of Bombs
Won a Multiplayer Battle.
Used 1000 Bombs
Over 5 Bombs Exploded as a Combo
1000 Blocks Destroyed
Maximized your blast radius
Won without getting a panel
Lost to a Revenge Cart
Created a Room and Battled
Got defeated by your own bomb.
Collected all Trophies
Used 1,000,000 Bombs
1,000,000 Blocks Destroyed