As you can tell , fail-safe has many different applications -- you need to determine what the fail-safe mode is for your application, determine what the failure modes are, and apply those methods to allow the fail-safe mode to function [i.e. we had main gas regulators supplying a major power plant fail open so that a plugged filter would not shut down the entire plant -- other provisions were incorporated to control the gas pressure; or the (steam turbine) Emergency Lube Oil Pump should fail to run (to failure, if need be) to keep the turbine bearings lubricated -- it should start upon other pump failures, even a failed pressure switch...]
some things should fail on, fail off, fail open, fail close -- depending upon the application; even if you purchase a piece of equipment that has a fail-safe function, you should still evaluate (preferably before purchase) to determine if it fits the fail-safe mode you need... [some applications may be dictated by code, so you should also determine what applies to your application, i.e. fire pump service]
In some respects, this involves a separate study of your systems to make sure proper fail-safe modes are adequately designed into your systems..
to answer your question, both N.O. and N.C. contacts have application -- which to use will depend upon what needs to be done in failure mode...