Do there exist elliptic curves over schemes which have all primes as residue characteristics? It's well known that there are no elliptic curves over Spec $\mathbb{Z}$, but it's unclear (to me at least) if the proof generalizes.
My question is: If $S$ is a connected scheme such that has every prime as a residue characteristic, then can there exist an elliptic curve over $S$?
 A: Another perspective: Let $K$ be any number field and let $E$ be any elliptic curve over $K$ with $j$-invariant in $\mathcal{O}_K$. Then there is an extension $L$ of $K$ such that $E \times_K L$ extends to an elliptic curve over $\mathcal{O}_L$. 
Proof: By properness of the stack $\overline{M}_{1,1}$, there is some $L$ so that we get a stable genus one curve with one marked point over $\mathrm{Spec}\ \mathcal{O}_L$. We just need to rule out the possibility that some fibers are nodal. If the fiber over a prime $\pi$ of $\mathcal{O}_L$ is nodal, than a computation with Tate curves shows that $v_{\pi}(j)<0$, contradicting that $j \in \mathcal{O}_L$. $\square$
I think it is worth bringing this up, as well as the CM solutions above, because there are many more integral $j$-invariants than there are $j$-invariants with CM.
A: More interesting is the question of which fields admit what sorts of elliptic curves having everywhere good reduction. For example, how about quadratic fields? Here are a couple of results.
Theorem (Rohrlich [1]) Let $K$ be an imaginary quadratic field and $j$ the invariant of some fixed isomorphism class of elliptic curves with complex multiplication by the ring of integers of $K$. Put $F=\mathbb Q(j)$ and $H=K(j)$. Then there is an elliptic curve with invariant $j$, defined over $F$, which has good reduction at every place of $F$ if and only if the discriminant $−d$ of $K$ is divisible by at least two primes congruent to 3 modulo 4. 
Theorem (Setzer [2]) There are no elliptic curves having good reduction everywhere over $\mathbb Q(\sqrt{-m})$ for $m=1,2,3,5,6,7,10,13,14,15,17,21,22,\ldots$ .


*

*Rohrlich, David E., Elliptic curves with good reduction everywhere.
J. London Math. Soc. (2) 25 (1982), no. 2, 216–222.

*Setzer, Bennett, Elliptic curves over complex quadratic fields.
Pacific J. Math. 74 (1978), no. 1, 235–250.

A: Yes, there can. Choose any elliptic curve $E$ over $\mathbb{Q}$ with potential good reduction (for instance, a curve with potential CM) and pass to a number field $K$ over which the reduction is everywhere good. Then $E_K$ extends to an elliptic curve over the ring of integers $\mathcal{O}_K$, and the latter has all primes as residue characteristics.
