Here is the full list of all 12 The Simpsons Arcade Game trophies - there are 7 bronze, 4 silver and 1 gold.
From stage 1, make it to the stage 3 bosses in Survival Mode with the Japanese ROM.
Complete the game on Expert difficulty in individual Quarters mode.
Complete the game in 30 minutes or less.
Earn 250 points (US) or 25,000 points (Japan) in a game.
Defeat the Wrestler without any player dying.
Complete the game four times, once with each character.
Hurt Smithers with one of his own bombs.
Playing from stage 1, get Bongo to appear in 4 different levels.
Playing from stage 1, defeat the battling robot.
While performing a Team Attack, damage 25 enemies.
Playing from stage 1, fall through the deck after Smithers.
Play in a 4-player match with three other people.