I think you want to look at Guillemin and Golubitsky's book Stable mappings and their singularities, which has a thorough description of what the singularity types of stable mappings are between surfaces.
Basically, the only stable singularities for smooth maps between surfaces , i.e., (i.e., $2$-manifolds 2$-manifolds) are folds and cusps, and these cannot usually perturbed away by small perturbations.
If you don't restrict to stable mappings or some similar class, the kinds of singularities that can occur can be extremely complicated, and I doubt that there is any workable classification.
