We consider the analytic structure of interfaces in several families of steady and unsteady Stokes flows, focusing on the formation of corners and cusps in the presence or absence of surfactants. An overview of flows where interfacial singularities have been identified or alleged to occur is presented. Two problems are considered in detail. First, numerical studies show that, under certain conditions, a surfactant causes the tip of a deforming bubble in extensional flow to develop high curvature leading to the ejection of a sheet or column of gas by means of tip streaming. Second, numerical simulations of the coalescence of viscous cylinders show that a transient cusp may spontaneously develop at a finite time during the unraveling of the interfaces, and this occurs in a flow that is driven exclusively by surface tension. Replacing the ambient inviscid fluid with a slightly viscous liquid suppresses the cusp formation.