  We complete the description of semistable models for modular curves
associated with maximal subgroups of $\mathrm{GL}_2 ({\mathbb F}_p )$ (for $p$
any prime, $p>5$). That is, in the new cases of non-split Cartan modular curves
and exceptional subgroups, we identify the irreducible components and
singularities of the reduction mod $p$, and the complete local rings at the
singularities. We review the case of split Cartan modular curves. This
description suffices for computing the group of connected components of the
fibre at $p$ of the N\'eron model of the Jacobian.
