They're not flash games but I have been playing a few browser games in the last week or so. I even subscribed to one (it was either that or let $10 credit on my pre-paid mobile phone expire...).
Solarfleet is the one I paid for, it's a 2D space mmorpg. You only get 500 turns on unpaid and they take 2 minutes to recharge so it's a little slow going. Hint: don't fly straight to your destination, take 5 turns at a time and blow the crap out of everything on the way.
Unification Wars is another space-based mmorpg. More text based (by that I mean it isn't animated... it does have graphics), but the ability to send a fleet of 5 million fighters 100 000 cruisers and 500 000 support craft to try and take over a system consisting of 20 000 planets is great fun. It's also a game you can play without ever having to pay, paying gives you a few more missions and that's it. There's even a way you can win a one-month paid subscription by finding relics. This is a good game and I highly recommend it.
I also played a bit of Imperia Online, but I havn't spent much time on that yet. Seems good.
There's another one I was playing which was decent, but the name escapes me at the moment.
I am currently downloading the 225mb client for Planeshift. This game looks amazing and I have been wanting to play it for about 18 months but I didn't have the OS version that could run it on Mac. Now I have a (stupid)PC I can finally play it. This is a 3D mmorpg along the lines of WoW and don't let the fact it's free scare you away. The quality of the graphics is superb. If you ever played RuneScape and remember the crappy graphics there, forget it. This is commercial quality stuff. I'll play it tomorrow and let people know if it's as good as it looks.
I know none of these are Flash, but I don't care - they're free.
I forget what the rules are for linking to things like this, especially Planeshift, so I'll just say to put the titles into Google...