Which multiplayer weapons are available at each rarity level?

You can see the multiplayer gun collection in-game. From the main multiplayer screen select "prestige", then "collection". Use the subtle top tab menu to switch from characters to weapons. Then use the filtering button until you get to the desired rarity.

Don't have time to look in game? Got you covered:

Common

Assault Rifle: Avenger
Pistol: Charger, Predator
Shotgun: Katana
Sniper Rifle: Viper

Uncommon

Assault Rifle: Cyclone, Mattock, Zalkin
Pistol: Carnifex, Phalanx
Shotgun: Disciple, Ruzad
Sniper Rifle: Incisor, Raptor, Widow

Rare

Assault Rifle: Hesh, Falcon, Revenant, Sandstorm, Thokin, Halberd
Pistol: Equalizer, Scorpion, Sidewinder, Talon, Hornet
Shotgun: Scattershot, Sweeper
Sniper Rifle: Indra, Vanquisher, Isharay, Lanat

Ultra Rare

Assault Rifle: N7 Valkyrie, P.A.W., Soned
Pistol: N7 Eagle, N7 Hurricane, Rozerad, Ushior
Shotgun: Dhan, N7 Crusader, N7 Piranha
Sniper Rifle: Naladen, Black Widow, N7 Valiant

Many items have incorrect categories in the collection menu. That's ME:A baby! I've corrected for this to the best of my knowledge.