Hi, and welcome to Using English.
The use of "at" or "in" are the same with supermarket as they are with theater, restaurant, stadium, etc.
If you say "at" it means anywhere in or immediately around it. If you say "in" you are referring specifically to the interior.
I would use "the" in most cases, even if the person I'm speaking with doesn't know (or care) which market I use. I would use "a" to ask something like "Can you tell me where there's a supermarket in town? I need to pick up some groceries."
(In fact, I use "grocery store" and not "supermarket" in my day-to-day vocabulary.)
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