  The event horizon of a dynamical black hole is generically a non-smooth
hypersurface. We classify the types of non-smooth structure that can arise on a
horizon that is smooth at late time. The classification includes creases,
corners and caustic points. We prove that creases and corners form spacelike
submanifolds of dimension $2,1$ and that caustic points form a set of dimension
at most $1$. We classify "perestroikas" of these structures, in which they
undergo a qualitative change at an instant of time. A crease perestroika gives
an exact local description of the event horizon near the "instant of merger" of
a generic black hole merger. Other crease perestroikas describe horizon
nucleation or collapse of a hole in a toroidal horizon. Caustic perestroikas,
in which a pair of caustic points either nucleate or annihilate, provide a
mechanism for creases to decay. We argue that properties of quantum
entanglement entropy suggest that creases might contribute to black hole
entropy. We explain that a "Gauss-Bonnet" term in the entropy is
non-topological on a non-smooth horizon, which invalidates previous arguments
against such a term.
