Most scientists do not think that black holes have a "real singularity." The idea of a mass that occupies a single point in space comes from taking Einstein's equations beyond the event horizon, ie. the curvature of space is greater than light so means nothing can have any resistance to distance/size. At some point between the event horizon and the center, quantum gravity (which is not understood) has to come into the picture. One idea is the "black hole firewall," which is that mass gets "stuck" at the boundary of the black hole.