  We study the Wigner caustic on shell of a Lagrangian submanifold L of affine
symplectic space. We present the physical motivation for studying singularities
of the Wigner caustic on shell and present its mathematical definition in terms
of a generating family. Because such a generating family is an odd deformation
of an odd function, we study simple singularities in the category of odd
functions and their odd versal deformations, applying these results to classify
the singularities of the Wigner caustic on shell, interpreting these
singularities in terms of the local geometry of L.
