It seems to be a folklore that for any genus $g$, there is a number field $K$ and a curve $X$ over $K$, such that $X$ has good reduction at all the places of $K$. Are any simple proofs of this?

One way of seeing this is by appealing to Rumely's general localglobal principle over $\bar{\mathbb{Z}}$, applied here to the moduli stack: an algebraic scheme over the algebraic integers $\bar{\mathbb{Z}}$ has a solution (point) in $\bar{\mathbb{Z}}$ if and only if it does in all $v$adic completions $\bar{\mathbb{Z}}_v$. I don't know if this is a simple proof, though  it is probably not what you are looking for. Here is a link to Rumely's paper: 

