We reexamine the focusing effect crucial to the theorems that predict the emergence of spacetime singularities and various results in the general theory of black holes in general relativity. Our investigation incorporates the fully nonlinear and dispersive nature of the underlying equations. We introduce and thoroughly explore the concept of versal unfolding (topological normal form) within the framework of the Newman–Penrose–Raychaudhuri system, the convergence-vorticity equations (notably the first and third Sachs optical equations), and the Oppenheimer–Snyder equation governing exactly spherical collapse. The findings lead to a novel dynamical depiction of spacetime singularities and black holes, exposing their continuous transformations into new topological configurations guided by the bifurcation diagrams associated with these problems.