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