Single use plastic bags are no longer used here in the UK either, but stronger 'bag for life' bags are used, and are replaced by the supermarket when no longer strong enough to carry items. Most people that I know use these bags, or bags that they have made themselves, baskets, fold up bags etc.
All countries should ban these single use bags, it is amazing how quickly you can get used to using other types of bag.
I use some fabric bags, and wash them when looking a little grubby.
Now, if only the supermarkets would cease wrapping every and any vegetable, in plastic that would be awesome. I try to buy loose veg, and take paper bags to put them in. Some supermarkets will be pleased to place your meat/fish purchases in your own containers too - wonderful.