The Class-Y noise suppression capacitors (normally 2.2nF) between mains side and secondary of the power supplies will pass a steady state AC current of approximately 166uA on 240V 50Hz mains. This current is way too small to cause any hazard to humans or computer operation.
There is another factor though. If you happen to plug the monitor when the mains AC voltage waveform is at its peak, that same 2.2nF capacitor will pass a very much larger instantaneous charging current, limited by many factors, mains wiring impedance between the computer and monitor sockets, their mains leads, the DVI or HDMI cable between them etc. In this particular situation might cause a crash depending on the computer's EMC susceptibility what it's doing at the time, or any number of other factors. It's impossibly to say other than 'might'.
This factor is which manufacturers of A/V equipment, computer equipment etc, warn against hot plugging of interconnecting cables. This spike current can occasionally be seen as a tiny spark between connector shells when not following this warning (166uA would never cause a visible spark).
Now you are asking whether plugging the monitor mains connection will cause a hang or crash. I think this is very unlikely, much more so than hot plugging the DVI/HDMI(/VGA?) cable, but it's impossible to give an absolute assurance.