  We establish several results combining discrete Morse theory and microlocal
sheaf theory in the setting of finite posets and simplicial complexes. Our
primary tool is a computationally tractable description of the bounded derived
category of sheaves on a poset with the Alexandrov topology. We prove that each
bounded complex of sheaves on a finite poset admits a unique (up to isomorphism
of complexes) minimal injective resolution, and we provide algorithms for
computing minimal injective resolution of an injective complex, as well as
several useful functors between derived categories of sheaves. For the constant
sheaf on a simplicial complex, we give asymptotically tight bounds on the
complexity of computing the minimal injective resolution using those
algorithms. Our main result is a novel definition of the discrete microsupport
of a bounded complex of sheaves on a finite poset. We detail several
foundational properties of the discrete microsupport, as well as a microlocal
generalization of the discrete homological Morse theorem and Morse
inequalities.
