All the weapons you need for this trophy are listed below so you can keep track as you play. The majority are either given to you at the start of one of the missions or are readily available to pick up from dead enemies. So the best advice is to check which weapons you have at the start of each mission, including the explosives that you equip using the d-pad, and make sure to get a kill with each one. You can start picking up every new weapon you come across, this was you'll use the majority of them and will only need to keep your eye out for a few remaining weapons.The launchers that lock onto vehicles only require you to destroy an occupied enemy vehicle for it to count.
The Mile High Club mission has a lot of enemies and they drop a wide variety of different weapons so if you’re not sure which ones you haven’t used yet and you’ve definitely done the 3 rare ones listed below as well as all the explosives, then try Mile High Club on Recruit and just farm all the weapons you come across. Like every other trophy in the game your progress toward this carries across between playthroughs and mission select and a kill will still count if you quit out of the mission immediately after.
There are 3 weapons that are rare and only attainable at specific points in the game though...
Desert Eagle - This is only dropped by one particular enemy during the Crew Expendable mission. In the first room with cargo containers in and the raised walkways on the right and left (same room as the second intel) an enemy will shout as he jumps out at you from inside or behind one of the containers. He will have the Desert Eagle. For more information see the Desert Storm trophy which involves getting 5 kills with it.
M16A4 - This one can only be found during the War Pig mission. Some of your allies will have it equipped so when any of them are killed check which gun they dropped.
M60E4 - This is only dropped by one particular enemy right at the end of the Game Over mission during the closing moments of the game. After you crash then climb up from the crumbled section of the bridge an enemy truck will drive toward you before reversing back up the road. The soldier that jumps out from the passenger side will drop the M60E4 when you kill him so run and grab it and use it on one of the other enemies while you have the chance.
Here is the full list of weapons in the game that you need to use:
GP-25 Grenade Launcher (possible attachment for the AK-47)
M203 Grenade Launcher (possible attachment for the M4A1)
An AK-47 with a grenade launcher can be found on the big weapons table in the first building you enter during Charlie Don't Surf, as well as in many other locations.