There are 12 Trophies that can be collected.

Bronze

Easier Than Arm Wrestling
Defeat the Wrestler without any player dying.

Falling For Smithers
Playing from stage 1, fall through the deck after Smithers.

It Tastes Like Burning
Hurt Smithers with one of his own bombs.

Dysfunctional Family
Play in a 4-player match with three other people.

Bongo's Angst
Playing from stage 1, get Bongo to appear in 4 different levels.

Battling Seizure Robots
Playing from stage 1, defeat the battling robot.

Helper Monkey
While performing a Team Attack, damage 25 enemies.

Silver

Mr. Sparkle
From stage 1, make it to the stage 3 bosses in Survival Mode with the Japanese ROM.

Best Score Ever
Earn 250 points (US) or 25,000 points (Japan) in a game.

Release The Hounds
Complete the game in 30 minutes or less.

Like Taking Candy From a Baby
Complete the game four times, once with each character.

Gold

"Excellent"
Complete the game on Expert difficulty in individual Quarters mode.

http://www.ps3trophies.com/forums/playstation-trophy-news/11424-simpsons-arcade-game-trophy-list.html