If there is a way to keep score, there is likely some kind of competition for it..
The ones I know about or are offered at my club:
NRA High power Rifle
Axis vs Allied, Bolt and Semi-auto
5-stand Sporting clays
Small bore target
Small bore Sillouette
Handgun Sillouette (IHMSA)
Black Powder Cartridge Rifle Sillouette
NRA Action Pistol
Um, that's all that comes to mind..
Had to add:
Air Rifle Sillouette