When they are attempting to serve you at work, and you are unavailable, can they speak to your supervisor?

That would be more up to your company policy than law. They can speak to anyone but that does not constitute serving you. They could, for example, ask when you might return or what shift you work. Most company supervisors would not respond.

My colleague, Ballenger, is 100% correct - as usual.

I own and operate a process service company. If I am attempting to serve someone at their place of employment I ask if I could speak to them about a personal matter. Some employers call them to the front office where I ask the person being served if we can speak privately. Other employers say no. Some tell me to leave the property (which I do).

I've served people at dinner in restaurants, for that matter.

It is my personal choice NOT to embarrass anyone. I know process servers who, quite frankly, don't much care. I would never go into a workplace with the summons in hand. I know process servers who do.

Depends on the law where you are, the Attorneys's instructions and your own gut feelings on the matter.