Although there's officially no Linux client, it's actually very easy to run Spotify from a Linux computer.
Spotify is a great application that allows you to stream music direct from Spotify's servers and play it on your local machine. There's a good Mac client, one for Windows, but as yet no Linux client.
However, it's actually quite quick and easy to get Spotify up and running on Linux when you know how. Spotify even provide details on how to do this on their website.
Download Spotify for Windows
First go to Spotify.com and download the Windows version of Spotify. This is a .exe executable.
We're going to run the Windows version of Spotify using the Wine software. Wine recreates a Windows-like environment within Linux allowing you to run Windows software relatively seamlessly. To install simply open a Terminal and paste in this command 'sudo aptitude install wine'.
Load Spotify with Wine Windows Program Loader
Find the Spotify.exe file in your Downloads folder, or wherever you saved it to, then right click it and select 'Open with Wine Windows Program Loader'.
Select install location
You can simply select the default install location when prompted. Wine doesn't save files to a place within your home directory so this doesn't really matter too much.
Login using your Spotify credentials
When the login form comes up, simply enter your Spotify username and password to login.
Play with Spotify