The way they've pretty much solved the plastic bag problem here (Austria) by charging for them. Result: ~99% of people bring (and reuse) their own shopping bags/baskets.

There are various incentives given by the UK supermarkets for using your own bags (IIRC, Tesco: "green" points, Sainsbury's: 1p/bag saved off your shopping, may have changed since I moved away from London) but they've got them the wrong way around, as there's no penalty for carrying on as before. If people start getting charged extra, a lot more will switch to using their own bags. With only positive incentive, my gf and I were the only people we ever saw who used their own bags.

The Austrian solution only applies to supermarkets, so other kinds of stores still hand out free bags, although I suppose supermarkets are (were) probably by far the biggest source of bags.