"Sport", "sports", and "sporting" as modifiers

Sport/Sports jacket is the same thing. It is a a suit jacket that is less informal than your normal business suit. In the midwest US I hear sports used more but when on the east coast I hear sport. The meaning is the same.

Sporting jacket is something that I have heard but it is not commonly used. If someone said they had a sporting jacket, the first thing that would come to my mind is hunting. It would bring up visions of men prancing around on their horses chasing foxes. Either way I wouldn't use this term in normal speech unless I was trying to convey hunting mixed with either upper-class or in some sort of dated story.