If you're in a store or restaurant and dissatisfied with the work of an employee and want to complain, how will you say:
Call your supervisor!
Call your manager!
or some other way?
Yes, in AmE, those seem fine. Although in certain contexts manager and supervisor are not interchangeable, I think that in the scenario described by OP, most people would not worry about the distinction. I've also heard people use boss instead of supervisor or manager.
I commonly hear
- Let me talk/speak to your ______.
- I want to talk/speak to your ______.
where you can use any of the names in the blanks.
(In BrE at least) Both may be used, although in my experience manager would be much more common.
It would also be common to ask for the manager if you were unsatisfied and the member of staff dealing with your complaint did not resolve it. This means that you are asking for the person currently in charge of the shop or restaurant, not just the supervisor/line manager of the particular staff member. In this context you would definitely not ask for "the supervisor", and if you did people might not understand what you meant.
A situation where the word supervisor might be more common would be if you were talking to someone on the phone in a call centre.
Based on what I know:
The word manager is a good a choice if wish to speak to someone who manages the restaurant.
I, personally, wouldn't use the word supervisor as it mostly addresses the manager of a manager, the person who supervises the work of a manager and assists him in directing and organizing the staff. However, it is a possible word too.
Be cautious and avoid slang and bad words when addressing the staff.
I think it has to do with the nature of the complaint. If it is about the service generally, ask the "the manager." If you are dissatisfied with a particular staff member specifically, ask for "your supervisor."