There are 36 trophies that can be collected.

Bronze

Artistic snobism
With Sly, silently replace a canvas

Gotta fine'em all!
Find 15 bottles containing a message

Bottled
Find 20 bottles containing a message

Thirsty
Find 30 bottles containing a message

Pink fits you so well!
Buy the Diable Fire Slam on the ThiefNet

That's really something!
Buy the Hover pack on the ThiefNet

Pink hell plate
Buy the Raging Inferno Flop on the ThiefNet

Thrifty
Buy the Paraglider on the ThiefNet

Fiery fox
Complete the prologue in Cairo by escaping Carmelita

Fox-trot lesson
Dance with Carmelita

Spider legs
Steal with Contessa thrice

Pickpocket
Get 100 coins

Emprisoned
Free Carmelita from the Contessa

Purse-snatching
Steal three keys from Carmelita in Canada

Vigorous hug
Force a bear attack in Canada

Heads or tails
Get 500 coins

Stowaway
Hijack the train

A friend in flames
Buy the Fists of Flames on the ThiefNet

Somebody called a medic?
Buy the Snooze Bomb on the ThiefNet

Ace of dodge
Buy the Combat Dodge on the ThiefNet

Reptile thief
Defeat Dimitri

Silver

Surfeit
Knock down Neyla to save Sly

Careful, it's spicy!
Defeat Rajan

A bad bug
Defeat the Contessa

A REALLY bad bug
Defeat the Contessa again

Exact money change
Get 1,000 coins

Digestive troubles
Find the Clockwerk's stomach

Gold medal
Defeat Jean Bison

Elk head
Steal the elk costume for Murray

Wake up!
Buy the Alarm Clock on the ThiefNet

Gold

True loot
Get 1,500 coins

Tick tock
Defeat Clockeyla

Remote control
Buy the Trigger Bomb on the ThiefNet

Stealth
Buy the Silent Obliteration on the ThiefNet

Ornithodentist
Get every bottle containing a message

Platinum

An unlikely bandit
Obtain every trophy in Sly 2: Band of Thieves

http://www.ps3trophies.org/game/the-sly-collection-sly-2/trophies/