Thanks to Nvidia’s Linux4Tegra and work of Bogdacutu and Steel01, as of today I’m running desktop Ubuntu on my Shield Tablet. And it works surprisingly well. Except some tablet-specific hardware like the modem, accelerometers and such, of course.
All it takes is to install MultiROM, patch the kernel and install the appropriate Linux4Tegra installation archive next to existing Android system and there you go: