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