  We construct noncommutative Donaldson-Thomas invariants associated with
abelian orbifold singularities by analysing the instanton contributions to a
six-dimensional topological gauge theory. The noncommutative deformation of
this gauge theory localizes on noncommutative instantons which can be
classified in terms of three-dimensional Young diagrams with a colouring of
boxes according to the orbifold group. We construct a moduli space for these
gauge field configurations which allows us to compute its virtual numbers via
the counting of representations of a quiver with relations. The quiver encodes
the instanton dynamics of the noncommutative gauge theory, and is associated to
the geometry of the singularity via the generalized McKay correspondence. The
index of BPS states which compute the noncommutative Donaldson-Thomas
invariants is realized via topological quantum mechanics based on the quiver
data. We illustrate these constructions with several explicit examples,
involving also higher rank Coulomb branch invariants and geometries with
compact divisors, and connect our approach with other ones in the literature.
