By an approach based on results of A. Ishii, we describe the versal deformation space of any reflexive module on the cone over the rational normal curve of degree m. To each component a resolution is given as the total space of a vector bundle on a Grassmannian. The vector bundle is a sum of copies of the cotangent bundle, the canonical sub-bundle, the dual of the canonical quotient bundle, and the trivial line bundle. Via an embedding in a trivial bundle, we obtain the components by projection. In particular we give equations for the minimal stratum in the Chern class filtration of the versal deformation space. We obtain a combinatorial description of the local deformation relation and a classification of the components. In particular we give a formula for the number of components.