Thomas Voß, one of the Canonical developers that has been working on Mir for months, tested out some other Linux desktop environments to see what would work on Mir if using XMir. XMir is the X.Org Server compatibility component, similar to Wayland's XWayland for being able to run legacy X.Org/X11 applications atop the modern display server.
Thomas was able to get the GNOME 3 Shell, Xfce, and LXDE running on XMir. With using XMir, the desktop environment's traditional window manager is running and communicating with XMir as it were an X.Org Server and is then talking to Mir as the system level compositor. The Mir compositor is basically treating the entire XMir/desktop as one child window. This isn't the most efficient process with in effect two compositors running, but it works, but again is similar to how XWayland works.
Thomas showed off the work on Google+ for Xfce, LXDE, and GNOME 3. Again though, this isn't any of the other non-Unity desktops running natively on Mir but rather through the XMir layer that provides X11 fall-back support so the traditional X window managers for the respective desktops are still in use and then XMir is rendering with the Mir compositor as if it were a single window.
The only Mir-native Linux desktop right now is Unity 8. By Ubuntu 13.10, Canonical wants Unity 8 with Mir as an experimental option on the desktop though for mobile Ubuntu Touch devices it may be in good standing then. Mir on the Linux desktop isn't expected to be stable until 2014. It was just last week that a package archive was created for Mir / Unity 8 for those wanting to try out this experimental code on Ubuntu Touch.