I use current-regulation on my nixie designs, and I've done cathode-side as well as anode-side regulation. Where you decide to regulate the current depends upon the driver you select. So far, all of my designs use direct-drive (non-multiplexed).
My 'Tomorrowland' clock uses HV5532 drivers (it has fourteen IN-18 tubes), so I regulate current on each anode. Yes, there are 14 current regulators.... My wristwatch uses NPN transistors to drive the cathodes, so I regulate the current on each segment.
Since 10mA fuses really aren't available, I size the resistor in the current-regulator so that it will operate at about 1/2 it's rated power during normal operation. For example, an 0805 SMT resistor is rated for 1/8 Watt, so I try to design for around 50-60mW. If a gross failure in the regulator occurs, the resistor should overheat and open. I have not tested it, though. I have a fuse on the power supply, and use the smallest value that will handle max operating current.
If you really want to be paranoid, monitor all of your anode currents with a multi-channel ADC. I've never done that, but it sounds like a fun project.
For cathode current-regulation, you would use an NPN or NMOS transistor; anode-side would use a PNP or PMOS. I found that high-voltage PMOS devices aren't nearly as common as NMOS.
If you use MOS devices, I'd recommend a zener-diode across gate-source and also very-high resistors (10Meg +) across drain-source to protect from ESD or unanticipated voltage spikes. It's probably overkill but SMT devices are so cheap and small it's worth the cost to me.