The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X_0(p)$ showing up in the cohomology of $X(p)$, whose associated Galois representations have unipotent local monodromy at $p$, which contradicts potentially good reduction.

$X(p)$ clearly does have good reduction at $p$ for $p=2,3,5$ because it has genus $0$.

$X(7)$ nonobviously has potential good reduction at $p=7$ where $X(7)$, the Klein quartic curve in $\mathbb P^2$, degenerates to a smooth hyperelliptic curve. See, for instance, Elkies's paper on the Klein quartic.

So $p=13$ is the only remaining case. What's the answer there?

The complete level modular curve $X(p)$ does not have potentially good reduction at $p$ for any $p \neq 2,3,5,7,13$ because then there are cusp forms on $X_0(p)$ showing up in the cohomology of $X(p)$, whose associated Galois representations have unipotent local monodromy at $p$, which contradicts potentially good reduction.

$X(p)$ clearly does have good reduction at $p$ for $p=2,3,5$ because it has genus $0$ and a rational point.

$X(7)$ nonobviously has potential good reduction at $p=7$ where $X(7)$, the Klein quartic curve in $\mathbb P^2$, degenerates to a smooth hyperelliptic curve. See, for instance, Elkies's paper on the Klein quartic.

So $p=13$ is the only remaining case. What's the answer there?

A good first step might be to try to answer the question for $X_1(13)$, which only has genus $2$.