Singularities of improper affine spheres and surfaces of constant   Gaussian curvature
  We study the equation for improper (parabolic) affine spheres from the view point of contact geometry and provide the generic classification of singularities appearing in geometric solutions to the equation as well as their duals. We also show the results for surfaces of constant Gaussian curvatureand for developable surfaces. In particular we confirm that generic singularities appearing in such a surface are just cuspidal edges and swallowtails. 
Go-o Ishikawa and Yoshinori Machida