Flash is supposed to be in the 2.2 release that should be out by the end of December. I have a Epic and for Flash I use a browser called Dolphin, there is also Skyfire, and both can be found in the Market and are free applications to download.
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I haven't been following the Flash issue too closely, but as I understand it, Flash 10.1 isn't officially supported on Android 2.1/Eclair. It is on 2.2/Froyo, and whenever we finally get that update, we should easily be able to install Flash from the market. I'm pretty sure I've seen it on AppBrain, so it's out there. I did come across this link -- http://www.addictivetips.com/mobile/install-flash-10-1-on-samsung-galaxy-s-on-android-2-1-eclair/ -- which I bookmarked for reference. These instructions are supposed to let you get Flash 10.1 onto a Galaxy S device running Android 2.1. I have not tried it myself, so I can't say whether it works or how good/useful this is. You could try this if it's important to get Flash working sooner rather than later. Just don't blame me if anything goes wrong. I hope this is of some help.
On my epic 4G with Froyo, I do have Flash 10.1 Installed, but firefox mobile browser doesn't seem to have a proper plugin, nor does adobe see firefox mobile as a legitamate platform for a plugin. As for using the built in browser, it's hard to say if flash 10.1 is working because most sites I go to, automatically load a "Mobile" optimized version.
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