  A minimal separator in a graph is an inclusion-minimal set of vertices that
separates some fixed pair of nonadjacent vertices. A graph class is said to be
tame if there exists a polynomial upper bound for the number of minimal
separators of every graph in the class, and feral if it contains arbitrarily
large graphs with exponentially many minimal separators. Building on recent
works of Gartland and Lokshtanov [SODA 2023] and Gajarsk\'y, Jaffke, Lima,
Novotn\'a, Pilipczuk, Rz\k{a}\.zewski, and Souza [arXiv, 2022], we show that
every graph class defined by a single forbidden induced minor or induced
topological minor is either tame or feral, and classify the two cases. This
leads to new graph classes in which Maximum Weight Independent Set and many
other problems are solvable in polynomial time. We complement the
classification results with polynomial-time recognition algorithms for the
maximal tame graph classes appearing in the obtained classifications.
