HSC will soon join the airwing and become one w/ HS, but for now, they arent' attached to an airwing. They operate on their own as detachments to replenishment ships and amphibs (for SAR). HCS is it's own weird animal, as it's a reserve component and does secret squirrel things on it's own. HSL is like HSC. It will eventually join HS and both will become HSM. For now, for the most part, they still operate out of their own squadrons and det out to small boys (frigates, cruisers, destroyers).
From what I've heard, HSC isn't too happy joining the airwing. HS isn't too happy joining up w/ HSL (HH-60 can correct me if I'm wrong), and HSL pretty much needs to join up w/ everyone else, or they'll never get to play.
As for the wings, there's a Carrier Air Wing, but there are also community specific wings. So a HS squadron may operate w/ the Air Wing 1, but they also belong to HSLANTWING or HSPACWING. They have a new name now, but you get the idea. HSL and HSC don't have the air wing component, so they only belong to HSLWINGPAC or LANT. Same same for HSC. Speaking of generalities here, as I'm sure there's exceptions (HSL-47, for example).
EDIT
Lowflier beat me to it. But one thing... HSM will do primarily ASUW along w/ ASW. Day to day, that's what the sea commanders use most.