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