Is graph connectivity definable in existential MSO with vertices and edges?
You can express that $s$ is connected to $t\ne s$ in $G=(V,E)$ by the $\exists\mathrm{MSO}_2$ formula


  “there exists $E'\subseteq E$ such that in the graph $(V,E')$, $s$ and $t$ have degree $1$, and all other vertices have degree $0$ or $2$.”


Note that the formula after the initial $\exists E'$ quantifier is first-order.

A graph of degree $\le2$ is a disjoint union of lines and cycles; since end-points of lines are exactly the vertices of degree $1$, the only way to satisfy the formula is that $E'$ consists of a line connecting $s$ to $t$, and possibly some disjoint cycles.

Likewise, directed connectivity can also be defined by an $\exists\mathrm{MSO}_2$ formula (we replace “degree $2$” with “in-degree and out-degree $1$”, and similarly for the end-points).