Hello Elizabeth,
Whilst available in the UK, Spotify is not as yet available here across the pond. If their audio loads into the browser (IE, Firefox, Opera or Chrome), try VideoCacheView from NirSoft -
http://www.nirsoft.net/; or, their WebVideoCap might allow you to actually capture the stream (both freeware).
Another UK site worth a look (one which does work here) is Last FM -
http://www.last.fm/ (though you'll run into similar issues downloading).
Cheers,
AJ