A linear stability analysis is performed for steady source‐vortex and sink‐vortex flows for viscous and inviscid incompressible fluids. The analysis is based on a method recently developed by the authors [Phys. Fluids7, 2345 (1995)], which utilizes the irrotational nature of the basic flow and takes vorticity as a perturbation. It is shown that source‐vortex flows are always unstable. Sink‐vortex flows are found stable except for low flow‐rate flows of viscous fluid. The potential vortex is unstable for three‐dimensional perturbations, but stable for two‐dimensional perturbations if the fluid is inviscid. For inviscid fluid the linear stability of doublet and higher‐order singularities for plane perturbations is also studied. The general integral of the linearized vorticity equations is found and used for stability analysis. Only doublet flow is found stable. A mathematical criterion for stability of certain types of steady two‐dimensional flows of inviscid incompressible fluid is formulated. ©1996 American Institute of Physics.