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