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.