Axiom's Market Basket LX is just the thing for running errands by bike. It's made with a tough steel frame and vinyl-coated steel mesh. It attaches to most racks (sold separately) with its adjustable, quick-release universal mount. And it boasts carrying handles so you can take it with you. You also get a place to easily attach a taillight (sold separately).
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the velcro strap broke the first time I tried to use it for shopping, it broke when I was putting it on the bike (empty), it didn't actually slide off the back of my bike getting it home, but there were moments when I needed to hang onto it. Will need something to keep it attached to the bike without sliding.