Multiplayer games are really rare in flash because of some complications that have held developers back. Generally speaking, while flash is an excellent platform for games because of it's web capabilities and ability to reach massive numbers of people, most flash developers are young and without finance. Running a multiplayer game requires a level of dedication and funding (servers) that your average 14-18 year old developer can't provide.
That being said, multiplayer flash games are on the up and up. Right now the most well known one is
Stick Arena. Personally I think the game is shit, but it's multiplayer, and it works. It's created and hosted by XGen, which is pretty much the top end of flash game development. They do some pretty serious projects over there.
On a significantly more indie "I made this in my bedroom after I finished my English homework" end of the spectrum, we just had a multiplayer racing game uploaded to Kongregate. The game is called Kongregate Racing, you can play it
here. Its not terribly impressive game play wise, but it was developed by only two people and it integrates some really cool Kongregate features like avatars and user names automatically. And since I work for Kongregate, I couldn't let a thread about flash games get away without mentioning it.
Also, there is the Mario MMORPG project that is flash based. You can check that out here:
http://xanthians.com/mario/index.php (took me a while to find that link again!) It looks like a really interesting project, but the game is a showcase for the limitations of flash as it preforms very, very, poorly. Not that it would be impossible to make a flash MMO, it would just need to be optimized very carefully to be playable, and the number of simultaneous users on screen would probably want to be kept low.
Hope those help you out!
(P.S. To shill my post up a little further, Kongregate is creating a multiplayer API for their website which will encourage and make multiplayer flash development easier by taking away some of the pain involved in multiplayer development (especially hosting and database storage), so I think that within a few months you'll see a lot more multiplayer flash games going on. I expect that API to be out within 2 months of now, most likely sooner.)
EDIT: Almost forgot to mention another XGen project that some friends of mine are working on called Stranded 2. I don't know what the estimated release of the game is, but it will have some multiplayer modes like deathmatch, ctf, team deathmatch. Similar to Stick Arena but a little slower and more aesthetically pleasing.
You can get the trailer for that here:
http://www.xgenstudios.com/play/stranded2The coder for that game is 16 by the way. Crazy ass kids and their internets.
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