Minor closed properties that are explicitly MSO expressible
I had an answer here involving apex graphs but it fails the definition of not having an explicit obstruction set given in this question: there is a published algorithm for finding the obstruction set, even though is too slow to run so we don't actually know what the obstruction set is.

So here's another parameterizable family of answers without that flaw (at least, as far as I know). Given a minor closed family $F$, and an integer $k\ge 1$, does the given graph $G$ have a $k$-ply covering graph in $F$? Much about this sort of question remains unknown: in particular, Negami's conjecture, which would characterize the graphs that have a planar covering graph, remains unproven. And it's minor-closed because any steps that you take to make a minor from $G$ can be copied in the cover.

To test for the existence of a $k$-ply cover of $G$ in $F$, in MSO$_2$, do the following steps:


Use the depth-first-search-tree trick to get an (arbitrary) orientation of $G$.
For each pair $(i,j)$ with $0\le i,j&lt;k$, choose a set of edges in $G$, supposedly the ones that have a covering edge that goes from ply $i$ to ply $j$.
Check that each vertex in each ply has exactly one copy of each incident edge, and thus that this information represents a valid covering graph of $G$.
Emulate the obstruction set based formula for membership in $F$ on the covering graph.