Here is the full list of all 51 Battlefield: Bad Company trophies - there are 42 bronze, 5 silver, 3 gold and 1 platinum.
Complete 'Ghost Town' on normal
Hit hostile helicopter with laser designator
Find all gold bars
Find all collectables
Complete 'Crash and Grab' on normal
Climb to the highest spot in the game
Complete one mission without dying (any difficulty)
Complete 'Air Force One' on normal
Kill 25 enemies in a helicopter
Destroy 50 vehicles. (containing enemy AI)
Drive all vehicle types (Jeep, Tank, Heli, Boat)
Complete 'Par for the Course' on normal
Find half of the gold bars
Find half of all collectables
Complete 'Crossing Over' on normal
Use Auto Injector 50 Times
Complete 'Acta Non Verba' on normal
Destroy 200 walls
Find one gold bar
Complete 'Welcome To Bad Company' on normal
(Online) Participate in one online match (Ranked)
Find 5 unique collectables
Kill 25 enemies
Knock down a small forest
Battlefield: Bad Company
Complete 'Welcome To Bad Company' on hard
Complete 'Acta Non Verba' on hard
Complete 'Crossing Over' on hard
Complete 'Par for the Course' on hard
Complete 'Air Force One' on hard
Complete 'Crash and Grab' on hard
Complete 'Ghost Town' on hard
(Online) Play 100 online matches
(Online) Destroy 1000 walls
(Online) Knock down 1000 trees
(Online) Use Squad in the menu, find a friend and play one round
(Online) Have 20 Squad members spawn on you without dying in a single round
(Online) Collect 5 unique dog tags
(Online) Collect 50 unique dog tags
(Online) Place #1 in a ranked match
(Online) Get at least one kill in every vehicle
(Online) Reach Rank 3
(Online) Reach Rank 8
(Online) Reach Rank 20
(Online) Reach Rank 25
(Online) 100 kills using vehicle
(Online) Take 3 pictures using the image system
(Online) Glide in the parachute for 3 seconds
(Online) Achieve all the awards
(Online) Achieve half of all awards
(Online) Get 10002 kills