There are 44 Trophies that can be collected.

Bronze

New Recruit
Complete the Unit 13 Training Course.

Snakechaser
Complete a Direct Action operation.

Lamplighter
Complete a Covert operation.

Speed Demon
Complete a Deadline operation.

L337 $0LD13R
Complete an Elite operation.

Less Me, More We
Complete an operation in co-op.

Friendly Rivalry
Get a higher mission score than someone on your friends list.

Adapt and Overcome
Complete a Daily Challenge operation.

By Strength and Guile
Reach level 10 with RINGO.

One Shot, One Kill
Reach level 10 with ALABAMA.

De Oppresso Liber
Reach level 10 with ANIMAL.

Honneur et Fidélité
Reach level 10 with CHUCKLES.

Facit Omnia Voluntas
Reach level 10 with ZEUS.

It Pays to be a Winner
Reach level 10 with PYTHON.

Trick Shot
Eliminate an enemy by setting off a mine.

Twin Foiled
Eliminate 2 enemies with a single melee attack.

Two For One
Eliminate 2 enemies with a single bullet.

Crowd Control Jr.
Eliminate 2 enemies with a single grenade.

Badass and Bulletproof
Eliminate 13 enemies without taking damage.

Brainstorm
Score 13 headshots in a single operation.

13 Squared
Eliminate 169 enemies over the course of your career.

Life of the Party
Show your enemy that you get down with the best of ‘em.

Blast from the Past
Find an homage to what has come before…

Extra Credit(S)
Watch the credits.

Silver

Variety is the Spice of Life
Complete 13 different Dynamic missions.

Action Hero
Complete all Direct Action operations.

Master Ninja
Complete all Covert operations.

Time Killer
Complete all Deadline operations.

Ch13f Op3r471v3
Complete all Elite operations.

World Peace in 36 Easy Steps
Complete all Covert, Deadline, Direct Action, and Elite operations.

Unstoppable
Achieve the highest score multiplier.

Lucky 13
Achieve a 5-star rating in 13 operations.

Dead Winger
Eliminate PHOENIX.

Witchiker
Eliminate WIZARD.

Anti-Venom
Eliminate VIPER.

Last Laugh
Eliminate HYENA.

Fat Chance
Eliminate BIG SLICK.

Sworded Out
Eliminate SCIMITAR.

Fangdango
Eliminate VAMPIRE.

Doublecrossed
Eliminate GRIFTER.

Finish Him!
Eliminate SCORPION.

Gold

Sine Labore Nihil
Max out all operatives.

Cloud Nine
Eliminate all High Value Targets.

Platinum

13-Star General
Obtain all Bronze, Silver, and Gold trophies in Unit 13.

http://www.ps3trophies.com/forums/playstation-trophy-news/11073-unit-13-trophy-list.html