Here is the full list of all 10 Super Bomberman R Online trophies - there are 7 bronze, 2 silver and 1 gold.
Dominate an area in a Battle 64
Died by destroying a Block
Died by your own bomb
Win a game without picking up a single item panel
Win a game while riding Louie
Come 1st in a Battle 64
Win a battle with all 8 Brothers
Raise firepower to the max
Raise the number of holdable bombs to the max
Raise movement speed to the max