There are some classical examples of such surfaces. For any prime number $p\ge 11$ different from 13, the modular curve $X_0(p)$ has good reduction away from $p$, and semi-stable reduction at $p$ (equal to the union of two projective lines intersecting at supersingular $j$'s). This is proved by Deligne-Rapoport (see also Bouw-Wewers: Stable reduction of modular curves, Prog. In Math. 224 (2004)).
There are however some contraints constraints for such curves over $\mathbb Z$. If $p$ is the unique semi-stable fiber, then Brumer-Kramer (Manuscripta Math.(2001)) showed that $p\ne 2, 3, 5, 7$, and Schoof (Compos. Math. (2005)) showed that $p\ne 13$.
