Squeezebox Server (formerly known as SqueezeCenter or SlimServer) has two ways of playing random music.
The "old-school" method of playing random music is to load the whole of a playlist into the current playlist with shuffle mode enabled, which will randomise the song order as the playlist is loaded.
Press the shuffle button to select either shuffle song or shuffle album.
Play something (such as all music, all music in a genre, or a playlist). SqueezeCenter will queue up all of the selected music, but randomised (either keeping all songs on an album together, or totally random).
This can have a performance hit for large music databases, as it will queue up all of the selected music as a single playlist.
Random Mix is the "new-school" method, which doesn't require the user to change the shuffle setting, and prevents performance problems by maintaining a selection of random songs in the current playlist.
It will keep a certain number of played songs in history, and you can see a bunch of future songs that have been selected for random play. The exact amount of songs selected and history are configurable (Web UI). When music is played, new music may be added to the end of the current playlist, and older (played) music will be removed from the start of the playlist.
From the player user interface, navigate to Plugins => Random Mix => Random Song Mix and press the Play button. That will clear any current playlist, and play the new selection of random songs (automatically adding new songs and removing old songs to manage the length of the playlist).
If you go back to Random Mix menu, you should see that it now says "Playing Random Songs" instead of the "Random Songs Mix" option. If you press play again, the current random playlist will be cleared and a new selection of random songs will be selected and played.
However, you can still press the "Add" button to add any other music to the end of the current playlist. So for example, you may be playing 1 of 11 random songs, and add an album to the end of the playlist. This would add items numbered for example 12..20. After that, random songs would start to be inserted onto the end again.
In the Plugins -> Random Mix menu, you could press Add on "Random Album Mix", and this would start to add random albums to the end of the current playlist.
At any time, "Play" will create a new current playlist, and "Add" will add to the current playlist.
Pressing the power button will retain the playlist and random play mode, so resuming playback would carry off where you left it. Stopping Slimserver (eg. powering off the PC that hosts slimserver) would lose the random play mode (although the current playlist will be retained).
Random Mix also allows specific genres to be selected. Only music that has a selected genre would be included in any random mixes.
There are third-party plugins available (such as "dynamic playlists"), which work in a similar way, but can be configured to select random music with more-specific rules, eg. "5 * rated music not played recently".