Start Songbird.
Type about: config in the location bar
Type useragent in the Filter box.
Right click on general.useragent.extra.songbird and copy the value to a text doc on your desktop for safe keeping. You may want to undo this change later!!!
Now modify that value to be Firefox/2.0.0.11. (Version probably not important.)
(Note: setting general.useragent.extra.songbird = Firefox/2.0.0.11 did not work for me.)
Now I can sign in to Rhapsody, but I'm prepared for other things to break because of that edit! If they do I'll change back to Songbird/0.4 (20071226135127).
When I have more time I'll try to investigate their javascript more thoroughly. And upgrade this from "Kludge" to "HowTo".
Reference: http://www.songbirdnest.com/node/2244