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?

1Perhaps you could just construct them; my guess is that something like $y^2=x^n+1$ has potentially good reduction at every prime. – Tim Dokchitser Aug 19 '13 at 8:05

Related: math.stackexchange.com/questions/1780629 – Watson Dec 3 at 13:29

As I mentioned in the above link, for $g=2$, an explicit example is given here. – Watson Dec 4 at 10:59
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:

1Rumely's paper solves the problem for affine irreducible schemes over the algebraic integers. The extension to Artin stacks is due to Laurent MoretBailly, see his paper Problèmes de Skolem sur les champs algébriques, Compositio Math. 125, 130, 2001. MoretBailly gives applications to moduli stacks of curves; see notably his Exemple 0.9. – ACL Aug 19 '13 at 7:34

3You can also apply Rumely's theorem directly to the moduli space of tricanonically embedded smooth curves of genus $g\geq2$: this is a scheme (subscheme of some Hilbert scheme), which is quasiprojective, smooth and surjective over $\mathrm{Spec}\:\mathbb{Z}$. Moreover it is a $GL_N$bundle over the stack $M_g$ for some $N$, hence has geometrically connected fibers over $\mathrm{Spec}\:\mathbb{Z}$ by DeligneMumford. So Rumely's theorem applies. – Laurent MoretBailly Aug 19 '13 at 14:52