Let $O$ be a Dedekind domain with fraction field $K$. Let $C$ be a smooth projective geometrically connected curve of genus $g>1$ over $K$.
Let $p:X \to \mathrm{Spec} \ O $ be the canonical model of $X$. Assuming that $C$ has semi-stable reduction over $O$, we have that $p$ is a stable curve.
When is $X$ regular?
One possible answer would be when the minimal regular model and the canonical model coincide.
But when does this happen? When there aren't any curves to be contracted on the minimal regular model. But I have a feeling this actually never happens...Why?