  We analyze the detection and classification of singularities of functions $f
= \chi_B$, where $B \subset \mathbb{R}^d$ and $d = 2,3$. It will be shown how
the set $\partial B$ can be extracted by a continuous shearlet transform
associated with compactly supported shearlets. Furthermore, if $\partial S$ is
a $d-1$ dimensional piecewise smooth manifold with $d=2$ or $3$, we will
classify smooth and non-smooth components of $\partial S$. This improves
previous results given for shearlet systems with a certain band-limited
generator, since the estimates we derive are uniform. Moreover, we will show
that our bounds are optimal. Along the way, we also obtain novel results on the
characterization of wavefront sets in $3$ dimensions by compactly supported
shearlets. Finally, geometric properties of $\partial S$ such as curvature are
described in terms of the continuous shearlet transform of $f$.
