Thanks for the reports - I was able to replicate the issue in Windows 8 and IE10 in the modern UI, until I installed all available updates for Windows - after which point the Web Player is working as normal. The updates listed included both flash player and IE updates.
Could you try making sure that your copy of Windows 8 is fully patched with all "important updates"? If not: could you install them, restart and let us know if that fixes the problem?
I was able to get around the issue by: Installing Fiefox from the Windows 8 Desktop, then go to the adobe website and install Flash. Since IE for Windows 8 and Chrome have Flash embedded I used FireFox. Using the link for alternate browsers should also work. It appears Spotify is using an older method to detect if flash is present on the box. This solution worked on three different machine/configurations over the weekend.
recently I joined spotify. I have a Mac 10.6.8 and ussually I use firefox, and I had instaled flash but this message keeps poping out 'To enjoy Spotify, please install Adobe Flash. It's free.' Tried Safari and same result.