Our local Tesco only does bags for life. One advantage of these is that, when they are too tatty to use any more, you can get them changed for free. Sainsburys and no doubt other supermarkets do this too. I order most of my shopping online and have opted for no carrier bags. Meat and items such as disinfectant and bleach usually come wrapped in a plastic bag for hygiene/safety reasons. You can even order fruit and vegetables as single items and these come in a paper carrier bag, which I find most useful.
I do, however, get most of my fruit and veg, and some meat, from an organic supplier. The fruit and veg usually come loose in a box, although some items are in plastic bags, which I reuse.
When my daughter and her family first moved from five minutes down the road to a place out in the sticks, the nearest supermarket was Asda - it was about a 30 mile round trip. She didn't like it but, for some items, seemed to have no choice. Eventually Tesco opened a small supermarket nearer to where she lived (bit bigger than a 'local' but a lot smaller than a 'metro'), but again not much choice. Where she lives now (still out in the sticks as far as I am concerned) actually has an enormous Sainsburys and the tiniest Tesco local that I have ever seen (although it does have an ATM, which saves at least another 5 minutes walk to the bank!).
As for me, I wouldn't do a food shop in Asda if they paid me. The nearest store never looks clean or tidy and doesn't have nearly enough checkouts or even trolleys to cope in peak shopping times, usually when people are going home from work. The last and only food item I ever bought there was an in-store baked loaf of bread 10 years ago. It was awful and went in the bin.