The surface of a liquid is easily deformed, stretched, and broken. Some examples are surface-tension driven break-up of a bubble, withdrawal and entrainment of stratified liquid layers and the splash of a liquid drop upon impact. Here we develop theoretical and simulation tools to analyze the dynamics and compare our results against experiments. Taken together, the results indicate that singularity formation can organize the evolution of the interface near a topological transition in distinctively different, and unexpected, ways. The asymptotic dynamics associated with the singularity needs not be universal, but can instead retain a memory of initial and boundary conditions.