Let$\newcommand{\mM}{\mathcal{M}}$ $\mM_{1,1}$ be the moduli stack of elliptic curves. Let $R$ be a Dedekind domain, say $\mathbb{Z}[1/N]$ for simplicity, and suppose we have a finite etale cover:

$$\mM\rightarrow\mM_{1,1}[1/N]$$ Must the coarse moduli scheme $M$ of $\mM$ be smooth over $\mathbb{Z}[1/N]$? This is certainly true if $\mM$ is representable, since $\mM_{1,1}$ is smooth over $\mathbb{Z}$. Furthermore, for a geometric point Spec $k\rightarrow$ Spec $R$ of characteristic not 2 or 3, $M_k$ is the coarse moduli scheme of $\mM_k$, which is normal since it's the quotient of the scheme $\mM\times_{\mM_{1,1}[1/N]} \mM(N^2)$ by $GL_2(\mathbb{Z}/N)$, where $\mM(N^2)$ is the representable stack over $\mathbb{Z}[1/N]$ classifying elliptic curves with full level $N$ structure. Since $M_k$ is a normal curve, it's smooth, and thus since $R$ is a Dedekind domain, we find that $M[1/6]$ is smooth over $\mathbb{Z}[1/6N]$.

The same argument doesn't work at $p = $ 2 or 3, since for a residue field $k$ of $R$ of characteristic 2 or 3, $M_k$ is not necessarily the coarse moduli scheme of $\mM_k$ (the stack is not tame over 2 and 3). Certainly the only potential singularities are the preimages of $j = 0\equiv 1728\mod p$ in $M_k$.

Is this a real obstruction? Of course if $\mM$ is representable then this is a non-issue, and maybe for special problems like $\Gamma_0(N)$ one can use division polynomials, but that's kind of ad hoc. I'd like to understand better what can go wrong generally speaking. If this is a real problem, then can we at least say it's flat?