G. Mark’s Answer
There were generally two types of supervision I've encountered. First was the supervision that was very hands-on and involved in what you did. This was great for new employees and small teams. The other was the supervision that was truly involved primarily in management issues, and they needed you to just take care of business yourself and tell them only if you needed them to intervene.
These two management styles are both valuable and necessary. You need to be aware of management's expectations and needs from you as well as the other way around.
A good thing to keep in mind is that management is not necessarily a two-tier situation -- you and the boss. Sometimes, depending on situations, it may be valuable for you to be a mentor or sub-supervisor or organizer or whatever you want to call it. You should try to run interference to shield your management from having to do too much meddling or putting out fires, so work with other team members to help them. You may need the same from them at any time.
One thing I've been bitten by is a drive to shield management too much. Not keeping an open channel of communication can cause them to be isolated and not informed of crises soon enough. The drive to be self-sufficient can sometimes devolve to being isolated and being a weak spot in the team. So strive to inform when necessary, ask for feedback when possible, take care of problems when practical. Keeping communication open but not overwhelming or a source of too much overhead is a good idea.