Are you part of US-GAF or something? I'd always dress smart/smart-casual to an interview, but a full suit is reserved only for career jobs. If I was hiring at Tescos and someone came in a suit, I'd worry about their ability to dress/function appropriately for the task at hand. Wouldn't make a snap judgement just off that fact, mind, but it wouldn't endear me to them.
Really? What should you go to Tesco in then, trackies and a t-shirt so you'll fit in? IMO a suit is a great all-purpose piece of smart clothing. Because if you've not got a suit then you've probably still wearing shoes, trousers, shirt and mayeb a tie. And without a jacket, a shirt looks a bit nekkid, especially sans-tie. How about a shirt and jumper combo, that could work. Would still look better with a jacket though.
I don't see how over-dressing for an interview could be seen as not being able to dress apropriately though - surely it would be an indicator that this person would indeed be able to stick to a dress code if it were laid out.
Really, I guess it depends how comfortable you are in a suit. I never used to like them but I'm into it now (wearing mine as I type), I wouldn't dream of wearing less to a future interview no matter where it was.