We tested pretty extensively in Firefox, but there's always things that can go wrong. Also if you think you have any info that will help us fix it, hit up the Contact Us link at the bottom of http://help.grooveshark.com/
Well, if I run FF from a different account it seems OK, so my guess is that the scripting is not playing well with Greasemonkey or Firebug or some other add-on.
So, on the one had, the trouble is at my end, but on the other hand most sites have no trouble with my add-on arrangements, so there's something that's at least a little sensitive in the GS scripting.
Follow up: Disabled add-ons to see if one was causing trouble, and it's Flashblock 1.5.13. Usually it shows a placeholder you can click to allow the Flash object to run, but it doesn't seem to show for me.
edit: just tried the site in Firefox ; can't even get the sign-in form to react. :(
Seems OK in Chrome, but at least the Flash version worked in FF for me.