Suppose $\mathfrak{F}$ is a finite collection of finite undirected simple graphs. Let’s call it minor-regular iff for any finite undirected simple graph $\Gamma$ it satisfies the condition: If $\Gamma$ contains a graph from $\mathfrak{F}$ as a minor, then it contains a graph from $\mathfrak{F}$ (maybe a different one) as a topological minor.
One-element minor-regular collections of graphs are already classified:
$\{\Gamma\}$ is minor-regular iff the degrees of all vertices $\Gamma$ do not exceed $3$
The proof that «the degrees of all vertices $\Gamma$ do not exceed $3$» implies «$\{\Gamma\}$ is minor-regular» can be found here. The inverse statement can be proved by contradiction using a construct from there.
My question is:
Does there exist some sort of classification of two-element minor-regular collections?
Such collections can be formed as unions of pairs of the above collections (as the class of all minor-regular collections is closed under finite unions), however, that is not all:
$\{K_5, K_{3, 3}\}$ is minor-regular
(a corollary of Wagner and Kuratowski theorems)
And there definitely should be something else…