Your special weapon is a weapon for special situations, like close quarter or if you run out of primary weapon ammo.

The Best Primary Weapon
The Best Heavy Weapon
All Destiny Guides

The Best Special Weapon PvP The Crucible Destiny

Shotguns are without question the best special weapon in Destiny. Whenever you are walking around in close quarters it is worth pulling out your shotgun. Whenever you shoot at someone be sure to melee straight afterwards to ensure you get the kill. The best shotguns are those with more range, and those that have a tighter spread when you aim down the sights.

Fusion rifles are very difficult to use and it is either better to use your primary or a shotgun instead of a fusion rifle in most situations. Sniper rifles can be fun but are not very competitive unless you are far away from enemies, which is rarely the case due the the map design in Destiny.