Do singularities have wave functions and in particular did the big bang singularity have one?

To be sure, many wave functions have singularities. In general, the wave equation has two families of independent solutions. One of those families exhibits no singularities. The other does, that is, it becomes unbounded or infinite at a particular point in space or time. The wave equation in a spherical coordinate system is a well understood and classic example. Along the radial coordinate, the solutions are Bessel Functions. There are two well studied familes of solutions. The J-Bessel Functions are well behaved everywhere, that is, they do not exhibit singularities. In contrast, the Y-Bessel function has a singularity at the origin or the coordinate system. The J and Y Bessel functions may be superimposed to form traveling waves. Depending upon how they are combined, the waves may travel away from the origin or towards the origin. There are many other coordinate systems besides the spherical one. They all have wave functions with and without singularities. If one of the coordinate systems conforms to our notion of the shape of the universe, then, with suitable boundary conditions, a singularity at the origin of time and space does indeed give rise to wave functions.