Have you tried this?
After playing the clip, stay online but minimize your browser. Go to your Windows Explorer and look in "Windows/Temporary Internet Files" and look for a file with an ".mp3" extension. (usually last file in the folder listing, but not always). Pick it up with your mouse (copying is preferable) and move it to a directory of your choice. When off line, click on it to see if it plays. If it plays, you're good. If so, you can rename it but leave file extension at mp3 or whatever. If it does not play, try giving it another file extension. Sometimes works,sometimes not.
This moving (or copying preferably ) it out of the Windows/Temporary Internet Files folder works for me usually with .mp3's and .ra's and some other music extensions. Some mini clips or full songs playable online, however, are only a ram or other streaming format so this won't always work.
You have to do it before shutting down, of course, and before closing your browser (if you have "delete tempory internet files" checked to auto delete. It mainly depends upon the "hurl" etc, the format of the clip or full song, and several other factors.
Worth a shot and harmless in my opinion as long as they're only for your own listening pleasure. I do it all the time.