How do I check my version of GNOME-Shell?

Just type

gnome-shell --version

into a terminal.

The gnome-shell man page doesn't show --version as an option. Assuming you have installed it from the ubuntu repositories then you can look up the version in the repository. To just show the version:

apt-cache show gnome-shell | grep Version

On Ubuntu Maverick (10.10) I get

Version: 2.31.5-2ubuntu2

Note that this works whether or not the package is installed. You can also look up all packaged versions of gnome-shell on all ubuntu versions on http://packages.ubuntu.com/