A crucial question in analyzing a concurrent system is to determine its long-run behaviour, and in particular, whether there are irreversible choices in its evolution, leading into parts of the reachability space from which there is no return to other parts. Casting this problem in the unifying framework of safe Petri nets, our previous work [3] has provided techniques for identifying attractors, i.e. terminal strongly connected components of the reachability space, whose attraction basins we wish to determine. Here, we provide a solution for the case of safe Petri nets. Our algorithm uses net unfoldings and provides a map of all of the system's configurations (concurrent executions) that act as cliff-edges, i.e. any maximal extension for those configurations lies in some basin that is considered fatal. The computation turns out to require only a relatively small prefix of the unfolding, just twice the depth of Esparza's complete prefix.