In this report we explore the local geometry of the medial axes (MA) and shocks (SH), and their structural changes under deformations, by viewing these symmetries as subsets of the symmetry set (SS) and present two results. First, we establish that the local form of the medial axes must generically be one of three cases: endpoints \math, interior points \math, and junctions \math. The local form of shocks is a sub-classification of these points.Second, we address the (classical) instability of the MA, i.e., abrupt changes in the representation with a slight changes in shape, as when a new branch appears with slight protrusion. The identification of these "transitions" is clearly crucial in robust object recognition. We show that for the medial axis only two such instabilities are possible: (i) when four branches come together \math, and (ii) when a new branch grows out of an existing one \math. Similarly, the six cases of shock instabilities are sub-classifications of these. The identification of these skeletal instabilities allows us to make equivalent structurally distinct skeletons arising from highly similar shapes, thus, regularizing the recognition process.