There are 60 Trophies that can be collected.

Bronze

Grandpa Got Served
Delivered the subpoeana to Arthur McFly.

Junior Bootlegger Badge
Managed to get 190-proof alcohol delivered to Emmett's.

Jailbreak!
Acquired Emmett's Rocket Drill.

Doc's Legacy
Got Doc's Notebook back from Biff.

Hello, Young Friend
Talked Emmett into completing the Rocket Drill.

A Plausible Explanation
Discovered the origin of the 'new' DeLorean.

Fuelamaster
Flawlessly mixed up a batch of Emmett's Rocket Fuel.

Fate In Flux
Heard about the curious fate of Marshall Strickland.

Hi Bob!
Namechecked 'Back to the Future' creators.

Deja Marty
Perfectly repeated dialog while repeating Doc's temporal experiment.

Escape Artist
Find a hidden passageway.

Police Blotter
Dry a cop out.

Moll Mender
Get a gangster's girl to do the right thing.

Tannen Tamer
Defeat the Tannen Brothers in six jumps.

A Minimum of Panic
Rescue Arthur with minimal fuss.

Art Lover
Track down a missing accountant.

Tuned Out
Enjoy a full set from Hill Valley's leading songstress.

Door Prize
Break the password code without cheating.

Chucklehead
Find 'Chuckles Lenart'.

Demystifier
Get a full explanation of your situation.

Hope You're Insured
Use the DeLorean parts to enter Hill Valley.

Intervention
Got caught with Lorraine's booze.

Seven Minutes in Heaven
Got caught making out with Jennifer.

All Good Dogs Go To Utah
Got caught with Einstein.

Into the Garbage Chute, McFly-boy
Jumped into the Decycling Bin.

Hill Valley Tourist
Listen to all tourist audio guides in Hill Valley.

One Bad Apple
Collect every Demerit in the game.

Marty and the Pinheads
Use the spray can to paint Marty's band name in the alley.

Hidden Treasure
Examine every container of contraband in the speakeasy.

Ruin it for Everyone
Coax Edna out of the courthouse with the buzzer.

Breaking Up is Hard to Do
Broke up Emmett and Edna.

Free Citizen Brown!
Rescued Citizen Brown from the Citizen Plus program.

The Power of Love?
Escaped from Marty's cell.

Let Them Eat, Um, Cake
Offered algae cakes to everyone.

Edna Was Right!
Revealed the Hill Valley hooliganism rate.

The Other Woman
Got Trixie ready to make Edna jealous.

Like a Not-So-Fine Wine
Messed with Emmett's cleanser.

A Not-So-Beautiful Mind
Crafted a mind map of Emmett that reveals him to be a 'Degenerate Criminal.'

An Uncomfortable Truth
Emmett is prophetically honest about his marriage prospects.

Timely Traveller
Take the fewest possible trips to Emmett's lab.

There's a Moose?
Get a good look at Trixie's postcard.

Emmett Finds His Muse
Emmett completes his spectacular failure at the Expo.

Crackpot Mended
Restored Edna's memory.

Raising the Bar
Prevented the Great Hill Valley Fire.

Green Gobblin'
Treated Desert Edna to an algae cake.

A Prickly Pair
Had a conversation with a cactus.

No Backups
Complete the glass house maze without going back up the stairs.

False Impressions
Impersonated three people over the phone.

Get it on Tape
Recorded everyone at the Expo.

Full Fair
Get your money's worth at the Expo.

Two-Up
Solve the saloon puzzle with no more than two trips to the second floor.

Moonwalker
Busted a move, 80's style.

Silver

A Bicycle Built for Two
Rescued Doc from Kid Tannen.

Back In Time
Traveled back in time to rescue Doc.

Kid Restraint
Defeat Kid Tannen.

Final Solution
Layout Biff and escape the speakeasy.

To the Expo!
Marty and Emmett head to the Expo.

Your Own Personal Frankenstein
Gave Emmett his Epiphany.

Gold

The Future isn't Written
Returned home.

Platinum

Platinum Trophy
Earn all trophies in: 'Back to the Future: The Game'

http://www.ps3trophies.com/forums/ps3-trophy-news/4766-back-future-game-trophy-list.html