What happens when a star undergoes gravitational collapse? Immediately prior to becoming a supernova the core of some types of stars may suffer gravitational collapse. 


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*What happens to any planets in orbit around the star at the instant the mass is fully collapsed? 

*Assuming this sudden change would cause some perturbation ; how large/distant would a planet in the system have to be to be relatively immune to such perturbation?

*Could unexplained perturbation serve as an indicator of a historic supernova in the vicinity?
 A: It depends on the nature of the system, and the explosion.  If more than about half the mass of the system is lost from the central star, the planet will become unbound (interesting National Geographic article on the subject).  This can be relevant even before the actually supernova - as massive stars lose a lot of mass through winds.  In any case, if the surviving remnant is massive enough, the planet will stay in orbit.
Nothing happens at the instant of collapse, because the planet doesn't 'know' about it until the changes in gravity become important.
The dynamic effects of supernovae ejecta can be important.  In particular, if the planet is a gas giant - significant amounts of mass can be blown off by the ejecta.  If I recall correctly, even close in rocky planets can absorb enough energy to become disrupted themselves.
There are some additional perturbative effects that are important, especially possible 'supernova kicks' (see for example this astrobites article).  Kicks are sometimes considered in binary star systems to explain large eccentricities (for example).
Currently it's hard to find planets around non-main-sequence stars (despite the first exoplanet being found orbiting a pulsar---which also shows that planets can remain bound after supernovae), so it's a bit early to seriously examine what types of thing to look for in those cases.
A: 
"What happens to any planets in orbit around the star" 

Why would anything happen to them? There is still a mass there; the same mass as before. It still has the same center of gravity. 
Now, when the radiation and shock waves arrive lots of stuff starts happening, including the effect mass around which the planets are orbiting dropping as material propagates outside of the planetary radius.

"at the instant of gravitational collapse" 

Remember that gravitation propagates at the speed of light, so even if anything where going to happen (it won't) it takes time for the information to get out to the planets.

"Assuming this sudden change would cause some perturbation"

It won't, so we don't assume that, which also disposes of your fourth point.
