I'm not an expert and this might be wrong, but I think that <a href="http://en.wikipedia.org/wiki/Cerf_theory">Cerf theory</a> should be impossible for orbifolds, and therefore all that comes from it, <em>e.g.</em> Kirby Calculus. Could somebody who knows please confirm this? I'm guessing that this comes <b>not</b> from orbifolds having singularities, <b>but rather</b> from orbifolds having built-in symmetries which mess up the necessary stratifications of the space of smooth functions to $\mathbb{R}$.

For example, <a href="http://en.wikipedia.org/wiki/Kirby_calculus">Kirby's Theorem</a> comes from the fact that a generic path between Morse functions in the space of smooth functions to $\mathbb{R}$ involves only finitely many which are not Morse-Smale (these correspond to handleslides). But I think that this is just wrong for smooth orbifolds. In the orbifold case, Morse functions satisfying the Morse-Smale condition (transversality between stable and unstable manifolds at a critical point) are not dense among smooth functions to $\mathbb{R}$, so perhaps a generic path between two Morse functions in the space of smooth functions might contain all kinds of craziness, and I doubt that there exists a sensible finite set of local moves between handle decompositions to parallel the Kirby moves.