I'm not an expert and this might be wrong, but I think that Cerf theory should be impossible for orbifolds, and therefore all that comes from it, e.g. Kirby Calculus. Could somebody who knows please confirm this? I'm guessing that this comes not from orbifolds having singularities, but rather from orbifolds having built-in symmetries which mess up the necessary stratifications of the space of smooth functions to $\mathbb{R}$.
For example, Kirby's Theorem 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.