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
Used 1,000,000 Bombs
1,000,000 Blocks Destroyed
Played 1,000 Multiplayer Battles
Played 100 Multiplayer Battles
Beat Beginner on Story Mode
Beat Veteran on Story Mode
Created a Room and Battled 100 times
Beat Expert on Story Mode
Created a Room and Battled 10 times
Won a battle with each of the 5 Dastardly Bombers
Won with a Revenge Cart in a 4 Player Multiplayer Battle
Created a Room and Battled
Beat Black Hole on Story Mode
Beat Planet Scrapheap on Story Mode
Beat Planet Lalaland on Story Mode
1000 Blocks Destroyed
Beat Planet Brainwave on Story Mode
Over 10 Bombs Exploded as a Combo
Beat Planet Timbertree on Story Mode
Used 1000 Bombs
Lost to a Revenge Cart
Tied with a Time Up
Won a battle with each of the 8 Bomberman Brothers
Won a Multiplayer Battle.
Won without getting a panel
Beat Planet Technopolis on Story Mode
Maximized your movement speed
Over 5 Bombs Exploded as a Combo
Maximized your blast radius
Carried the MAX number of Bombs
Got defeated by your own bomb.