SmartFoxServer is a good choice. I played around with it a few years ago with AS2, so might be work a look.
They offer a free lite version, but it does not support AS3. I think you need to get the basic plan for that. The pricing plans do look well priced though.
They have some nice documentation and plenty of example.
SmartFoxServer seems to be targeted at multi-player games, but it could be used for your needs.
EDIT*
Just found these Beta AS3 classes for SmartFoxServer. I haven't tried it out for my self though.
EDIT 2*
Another option would be to make your own. It's probably easier than you think if you have a few programming languages under your belt. Of course, this is the kinda thing that can suck your time away.
There are a bunch of tutorials out there for building your own socket server, and some are even specifically for flash. Here are some of the best:
I would definitely suggest using C# .net if you have Windows hosting. It would definitely be the simplest to build and would have the best performance. I would stay away from PHP for anything bigger than a chat app, but if that's all you have then go for it.