  We characterise subcategories of semistable modules for noncommutative
minimal models of compound Du Val singularities, including the non-isolated
case. We find that the stability is controlled by an infinite polyhedral fan
that stems from silting theory, and which can be computed from the Dynkin
diagram combinatorics of the minimal models found in the work of Iyama--Wemyss.
In the isolated case, we moreover find an explicit description of the
deformation theory of the stable modules in terms of factors of the
endomorphism algebras of 2-term tilting complexes. To obtain these results we
generalise a correspondence between 2-term silting theory and stability, which
is known to hold for finite dimensional algebras, to the much broader setting
of algebras over a complete local Noetherian base ring.
