Streamline patterns and their bifurcations in two-dimensional incompressible flow are investigated from a topological point of view. The velocity field is expanded at a point in the fluid, and the expansion coefficients are considered as bifurcation parameters. A series of nonlinear coordinate changes results in a much simplified system of differential equations for the streamlines (a normal form) encapsulating all the features of the original system. From this, we obtain a complete description of bifurcations up to codimension three close to a simple linear degeneracy. As a special case we develop the theory for flows with reflectional symmetry. We show that all the patterns obtained can be realized in steady Navier–Stokes or Stokes flow, but an unresolved difficulty arises in the symmetric case for Navier–Stokes flow. The theory is applied to the patterns and bifurcations found numerically in two recent studies of Stokes flow in confined domains. ©1999 American Institute of Physics.