  In this paper we generalize Artin-Verdier, Esnault and Wunram construction of
McKay correspondence to arbitrary Gorenstein surface singularities. The key
idea is the definition and a systematic use of a degeneracy module, which is an
enhancement of the first Chern class construction via a degeneracy locus. We
study also deformation and moduli questions. Among our main result we quote: a
full classification of special reflexive MCM modules on normal Gorenstein
surface singularities in terms of divisorial valuations centered at the
singularity, a first Chern class determination at an adequate resolution of
singularities, construction of moduli spaces of special reflexive modules, a
complete classification of Gorenstein normal surface singularities in
representation types, and a study on the deformation theory of MCM modules and
its interaction with their pullbacks at resolutions. For the proof of these
theorems we prove several isomorphisms between different deformation functors
that we expect that will be useful in further work.
