The topological description of flows in the vicinity of a solid bounda
ry, that is familiar from the aerodynamics literature, has recently be
en extended to the case of flow at a liquid-gas interface or a free su
rface by Lugt [Phys. Fluids 30, 3647 (1987)]. Lugt's work is revisited
in a more general setting, including nonconstant curvature of the int
erface and gradients of surface tension, using tools of modern nonline
ar dynamics. Bifurcations of the flow pattern occur at degenerate conf
igurations. Using the theory of unfolding, this paper gives a complete
description of the bifurcations that depend on terms up to the second
order. The general theory of this paper is applied to the topology of
streamlines during the breaking of a wave and to the flow below a sta
gnant surface film.