Generally speaking, any class could if they wanted to use any weapon that wasn’t locked to a certain profession, breed or side.
Your best if somewhat aged resource for what works and doesn’t work can be found in here:
Each profession has some sort of stickied guide that discusses weapons; some of it is dated, granted. (This keeper guide is fresh.)
Then there’s the fact that certain professions either don’t have weapon support (doctors famously mess around with weapon setups, chasing damage in exotic ways) or have support for several weapon lines (adventurer, soldier and enforcer most obviously)… and of course AO has a coprophile’s heaven in the form of weapons and armor. By way of example, this is a list of two-handed edged weapons with Fast Attack special attacks.
The trend, though, is:
Adventurer: 1h Edged and Pistol
Agent: Rifle (traditionally – MA agents are happening right now)
Bureaucrat: Pistol/shotgun (but really, it’s their nano skills)
Doctor: Pistol/shotgun(/MA/heavy weapons/piercing/1h Edged – you get the idea)
Enforcer: Any and all melee weapons, though Melee Energy isn’t supported as well
Engineer: Pistol/Grenade (their bots are their main weapon)
Keeper: 2h Edged (Melee Energy works these days too)
MA: Hand to hand. No weapons, but have combined attacks with some weapons (there’s the rare bow MA too, but…)
MP: Pistol/bow/2h Blunt/1h Blunt (pets are their main weapons)
NT: Non-expansion ones I’ve seen prefer pistol
Soldier: Any and all ranged weapons except bow with varying levels of support