Simple answer: It was my SD card, specifically a SanDisk Class 10 16gb model number SDSDU-016G-U46.
The long and the short of it is, not all SD cards are compatible with every operating system available for the RPi. In this case, the only media centre I could successfully install (and keep running after I rebooted) was Raspbmc. However other makes and classes of card did not have a problem.
A list of different SD cards and their success rates is available from eLinux, although this didn’t help me much, as I guess everyone is still figuring out what works with what.