My shift register is running with shared ground.
When I say "running" I mean the nixie lights up and changes and the arduino hasn't caught fire. It isn't doing what I expected in the slightest though.
Edit: Working properly now. I'd not paid enough attention again. Missed one of the ground pins for the 595 and had hooked the 74141 up to shift register pins 1-4 instead of 0-3.
Time to add a second nixie.