At least in Europe, since the introduction of the bag surcharge, the new plastic bags are thicker and much stronger. At most, they get punctured, but I have never had one break open on me.
Downside is that they are weak enough to break if mishandled.
Bigger bags to hold all your purchases - made from recycled plastic - are available from the cashiers for 20 cents.
Edit: Regular small bags are also available free of charge.