So I rode over with my Tikit. Folded it quickly and threw it into a shopping cart. I have a large Tikit, but it still fits in a normal sized cart without issue. There is even loads of room around teh folded bike to put food in the cart! Plus of course the lower shelf for big items.
Nobody cared I had a bike in the store with me - in fact nobody even seemed to notice! Using a backpack and a single front pannier I was able to haul quite a bit home.
As noted below the single pannier upfront is not ideal for hauling really heavy items. You can do it, but the bike wants to turn right all the time. For a short trip that's fine, but I'd get fairly irritated dealing with that for over an hour.
Bike Friday if you're listening the front rack is great, but we need one that can carry a pannier on each side.