The stability and bifurcations associated with the loss of azimuthal symmetry of planar flows of a viscous incompressible fluid, such as vortex-source and Jeffery–Hamel flows, are studied by employing linear, weakly nonlinear and fully nonlinear analyses, and features of new solutions are explained. We address here steady self-similar solutions of the Navier–Stokes equations and their stability to spatially developing disturbances. By considering bifurcations of a potential vortex-source flow, we find secondary solutions. They include asymmetric vortices which are generalizations of the classical point vortex to vortical flows with non-axisymmetric vorticity distributions. Another class of solutions we report relates to transition trajectories that connect new bifurcation-produced solutions with the primary ones. Such solutions provide far-field asymptotes for a number of jet-like flows. In particular, we consider a flow which is a combination of a jet and a sink, a tripolar jet, a jet emerging from a slit in a plane wall, a jet emerging from a plane channel and the reattachment phenomenon in the Jeffery–Hamel flow in divergent channels.