Just thought I'd post this as this issue seems to affect quite a few people and finding a solution on here can be pretty difficult. This solution worked for me on a standard Windows 7 PC. So here goes....
You launch Spotify in the normal way but it doesn't appear. You get the "egg-timer" for a few seconds but nothing else happens. You open Task Manager and you see that one or more instances of Spotify are running but you can't find them anywhere else.
You try uninstalling but that doesn't help. You try reinstalling but that doesn't help either.....
The reason uninstalling and / or reinstalling won't work is because both of them involve launching the app itself and because the PC can't see anything wrong with the way the program is running, they never get past that stage.
The Solution (hey, it worked for me!)
Shutdown your PC and then restart it in Safe Mode (hit the F8 key repeatedly during the boot up process then select Safe Mode from the menu that appears)
Once launched in Safe Mode, go to Control Panel, then Programs and Features, then find Spotify in the list and uninstall it (right click on it then click Uninstall)
Shutdown your PC and boot it back up as usual
Go to the Spotify website, download the installer and reinstall.
Spotify should now work.... if it doesn't you have at least ruled out a corrupted installation. If it does work.... You're welcome!!