You can work anywhere, though (new law alert!) until you are 18 it must include some form of education or training, or you must be still in education and only working part-time.
A supermarket is very possible - David S mentions Waitrose, and I know they do, including as checkout operators as well as shelf fillers. I live across the road from a Waitrose, so I shop in it a lot because it's as good as a convenience store or corner shop to me. If I want a bottle of wine, sometimes the checkout operator will call a supervisor to authorise it. Then I KNOW they're under 18, because if you're 16 or 17 you can't legally sell alcohol, as well as not being able to buy it. Such is the wonder of electronic tills... your age is programmed into it and it just won't allow the sale if you're not 18. For the same reason, the one thing they can't do with you is put you on the tobacco/lottery counter. But everything else is possible in a supermarket.
You are now legally required to remain in some for of education or training until the summer after your 17th birthday (it went up from 16 last year, and goes up to 18 next year) - so you can only get a job if it is recognised as including a significant element of traininf or apprenticeship.
Yes most of the larger supermarkets employ young people of 16 and over as shelf fillers. Waitrose pay the best wages
Yes you can work part time weekends and holidays, but the supermarkets will want you to be 16. Talk to them.
Apprenticeship boiiiiiii
Watch out for low wages!
