The Spotify tray (notification area) icon is well... useless and ugly, so I've removed it from my notification area and I though I'd share the "trick". Please note that I'm using the WINE (Windows) version of Spotify (so it might or might not work with the native Spotify client for Linux which you can currently use only if you're a paying subscriber).
Method 1 - removing only the Spotify tray icon1. Create a script anywhere you want on your system and paste this:
#!/bin/bash exec ~/.wine/drive_c/Program\ Files/Spotify/spotify.exe & sleep 10 killall explorer.exe exit
Then save the file.
Note that the "exec" line in the script above depends on where you've installed Spotify. I've used the standard path but if it's different for you, edit that line. If the Spotify tray icon is not removed, try altering the "sleep 10" value.
What the script does is: starts Spotify, waits for 10 seconds and then kills explorer.exe which allows Spotify to keep running but removes the notification area icon.
Important: you will now have to close Spotify by going to File > Exit not by clicking the close button (but the close button didn't close Spotify before this "hack" anyway - it only minimized it to the tray).
2. Make the script executable:
chmod +x /path/to/spotify
Where "spotify" is the script filename you've created and /path/to/ is... the path to the script.
3. You can now either edit the Spotify .desktop file or create a new launcher on your panel pointing to the script you've created and use that to start Spotify.
Method 2 - removing the whole notification area
WebUpd8 reader MastroPino sent as this tip: if you completely remove the notification area from the Gnome panel, the Spotify tray icon will be displayed in a pop-up. To remove this popup you can use TaskbarEliminator which even tough is a Windows applications, it does its job under Linux (using WINE) too.
Note that TaskbarEliminator doesn't work if you still have the notification area applet added to the Gnome panel.
Thanks to MastroPino for the second method!