There isn't any software serial number, so I assume you're talking about the hardware platform.
Most people are using HP/Compaq hardware. For these platforms, you want to use the HP Diagnostics utility. From remote desktop or VNC, login as Administrator and start the utility. On recent OS patchlevels, can find it in an "HP Information Center" folder at the top of the Start Menu. Go there and run "Diagnostics for Windows". Go to the Categories menu, then Asset control. Pick the "More" button at the bottom of the window, and the machine's serial number should be displayed.
IBM platforms have a similar utility that should show you the serial number, but I don't have access to one at the moment to give you a walkthrough.
