Under Windows:
Open Songbird and paste "http://fpdownload.macromedia.com/get/flashplayer/xpi/current/flashplayer-win.xpi" in the location bar at the top of the library or browser. Follow the installation process.

Under Linux:
Flash on Linux got a whole lot better with Flash Player 9. Go to the Macromedia site, and download Flash 9 for Linux. Decompress the .tar.gz file. If you run the installer - it will install the plugin for supported web browsers (Firefox, Mozilla, etc.), but it will not auto-detect Songbird. To install it for Songbird, copy the libflashplayer.so to your Songbird application folder's <tt>plugins</tt> directory. So if you have Songbird installed in ~/Songbird, copy it to ~/Songbird/plugins. Similarly, copy flashplayer.xpt to your Songbird application folder's <tt>components</tt> directory. Restart Songbird, and you should be back in business!