Before we start: this tip is only applicable to NTFS filesystems, and Windows 2000/XP/etc.
To pick a real example, liveroom.tv is a site that entirely written in Flash, so working out where the data is being streamed from is as inconvenient as going to the site and streaming lots of data every time you want to listen. However, as you watch the video stream into a buffer, there's a file in your user temp directory growing in size to several tens of megabytes.
Since the temp file is cleared when the video finishes playing, what you want to do is start it playing, then pause playback and wait for the video to completely buffer. Then get its filename and path — probably something like C:\Documents and Settings\[username]\Local Settings\Temp\fla11C.tmp
Because it's an open file XP won't let you copy it directly, and closing the Flash plugin (or forcibly modifying its grip with Unlocker) releases the file lock and wipes the file. However, recent versions of Windows contain VSS (Volume Shadow Copy Service) functionality, which a lot of backup software (Macrium Reflect was where I first encountered, but a lot of packages utilise it, particularly commercial ones.) For this example we'll use a bit of freeware called ShadowCopy produced by StorageQuest; whether you need to backup an open database or email mailbox, or want to raid your temp directory for buffered data, it's equally useful.
You'll want to give the video an .flv extension (it's an MPEG-4 variant, but won't work in Media Player Classic out of the box) and for playback I'd suggest VLC — if you want to get audio out of the file try FLV Extract then mp3DirectCut can be used to split up live sets into individual songs.
A bit more off-topic, but just in case anyone doesn't know this: for audio streams where you can get the direct URL, such as the RealMedia ones the BBC use, try an old freeware version of NetTransport. Works with rtsp:// but you might need a more recent nagware version of the app for other protocols. And for MySpace audio — though it's only 96kbps, remember — check out the link extractor at http://file2hd.com/