Do the risk assessment:
There is a school of thought which holds that a rotating machine must have an emergency stop function, though it doesn't have to be a big red button. It's actually written into the Machinery Directive Essential Requirements for Safety, but why?
In an industrial setting, a machine (pump, press, or whatever) will have to be guarded, and any operator access to rotating parts is prevented. This machine isn't made safer by fitting an E Stop in normal daily use, unless someone defeats the guards or otherwise removes the means provided to make sure that the operator (including casual passers-by) cannot access the bitey bits. That's not normal daily use, but it is a foreseeable hazard and the designer must take account of it. For this class of use the hazard comes from the VSD losing touch with reality, and going out of control. Unlikely, but possible and it's a high danger event however unlikely. So the designer must fit control measures (for example an E Stop) to permit the situation to be brought back under control. In this situation the best way of bringing the machine to a safe stop is to remove the mains supply from the VSD, and it doesn't matter a row of beans whether the VSD survives the experience or no. It's already compromised, switch the damn thing off any how you can.
The same isn't true for a lathe or especially (in spades) for a mill. Here the operator has to have access to the nasty dangerous bit to be able to do the task for which the machine is designed. The possibility of the VSD going doolally still exists, but there is a far higher probability of the operator being the root of all evil, and getting caught in the mangle. In this case the machine must be brought to a safe condition as quickly as possible, so using the VSD dynamic braking effect to slow the machine as quickly as possible is likely to be the proper control measure. That implies using the control stop function of the VSD, and definitely not removing the supply from the drive until the machine has been brought to rest.
There is a remnant hazard of the VSD losing the plot, just as there was with the industrial situation. So one could argue that there should be an additional control measure to cope with this as well. But one couldn't safely have two E Stop buttons between which the operator (with his hand in the mangle) has to choose, that's a silly idea, and furthermore you can't do both – i.e it's not going to work to have both E Stop methods working together, as they are mutually exclusive. So a lathe or a milling m/c (also a pillar drill or its close cousins) should have an E Stop function that invokes a braking effect, and does not isolate the machine. We just have to accept the remnant risk of the VSD going off on one. All the more reason not to skimp on the reliability and specification of the drive as installed.
Of course the braking effect must not of itself introduce another hazard worse that the one it is designed to control – eg unscrewing the chuck off the nose of the mandrel.
I've widely encountered a belief in industry that an emergency stop button is there so it can be "locked off" while someone is working on the machine. The requirements for safe isolation of the machine are NOT met by doing this, only an approved isolator (with the facility for a proper padlock and minimum contact separation, also clear indication of its having reached the safe isolation position) provides for safe isolation. In a home workshop single phase machinery is unlikely to have an isolator, though one would expect to be able to unplug it. This of itself isn't sufficient as it's too easy to plug the thing back in, perhaps unthinkingly or meaning to plug in another appliance. It wouldn't be sufficient in industry, but simply wrapping a few turns of insulation tape around the pins of the pug provides a memory prompt. Of course if the machine is of a stature it is hardwired to the supply it would necessarily have a local isolator, though domestic isolators (eg cooker switches) are usually not lockable and are therefore not isolators within the proper meaning of the term.
To sum up, my belief is that the best safety for a lathe. mill or drill in the home workshop is almost certainly to design a control system that uses the VSD to bring the machine to a controlled stop as quickly as possible. That implies not isolating the drive from the supply, and definitely not isolating the motor from the drive. Invoke whatever braking effect is safe on the way to a standstill.
Hope this helps
Simon