There are 45 Trophies that can be collected.

Bronze

Elite Ops
Complete a Story Campaign mission on Elite Difficulty

N00b
Complete a Competitive Multiplayer game

Good Game
Complete a Competitive Multiplayer game of every gametype

Combined Assault
Complete a Custom Campaign mission in Co-op

Extra Credit
Complete a Custom Campaign mission in Single-player

Government Spending
Call in a total of 10 airstrikes during your career

Death From Above
Eliminate 5 enemies with a single airstrike

Repo Man
Destroy a total of 50 vehicles during your career

Unseen Strike
Stealth kill a total of 10 enemies in a single mission

Grenadier
Eliminate a total of 25 enemies with explosives during your career

Crowd Control
Eliminate 3 enemies with a single explosive

Century Slayer
Eliminate a total of 100 enemies during your career

Melon Hunter
Eliminate a total of 100 enemies with a headshot during your career

Hat Trick
Eliminate 3 enemies in a row with headshots

Vorpal
Eliminate a total of 25 enemies with melee attacks during your career

Combat Medic
Execute a total of 25 teammate revives during your career

Armed to the Teeth
Unlock all weapons for use in Campaign Mode

I Call Her Vera
Master a weapon in any mode

Oracle
Collect an Intel item in a Story Campaign mission

Silver

Covert Ops
As Forty-Five, successfully maintain stealth in a Story Campaign mission

Over Achiever
Complete a Custom Campaign mission on every map in either Single-player or Co-op

Millennial Slayer
Eliminate a total of 1000 enemies during your career

Expert Tactician
Complete a Story Campaign mission using only your teammates to eliminate enemies

Information Warfare
Collect all Intel in the Story Campaign

Gold

Best of the Best
Complete the Story Campaign on Elite Difficulty

Above and Beyond
Complete a total of 100 objectives during your career

Shiny!
Master a weapon of each class in any mode

The Elite 45th
As Forty-Five, stealth kill a total of 45 enemies in the Story Campaign

Platinum

All is Said, All is Done
Obtain all Bronze, Silver, and Gold Trophies in SOCOM 4

16 Secret

http://www.ps3trophies.com/forums/trophy-news/56505-socom-4-trophy-list.html