  We study geometric and topological properties of singularities on the
boundaries of $\varepsilon$-neighbourhoods $E_\varepsilon = \{x \in
\mathbb{R}^2 : \textrm{dist}(x, E) \leq \varepsilon \}$ of planar sets $E
\subset \mathbb{R}^2$. We develop a novel technique for analysing the boundary
and obtain, for a compact set $E$ and $\varepsilon > 0$, a classification of
singularities (i.e. non-smooth points) on $\partial E_\varepsilon$ into eight
categories. We also show that the set of singularities is either countable or
the disjoint union of a countable set and a closed, totally disconnected,
nowhere dense set.
