We consider streamline patterns associated with single and double von Kármán point vortex streets on the surface of a nonrotating sphere, with and without pole vortices. The full family of streamline patterns are identified and the topological bifurcations from one pattern to another are depicted as a function of latitude and pole strength. The process involves first finding appropriate vortex strengths so that the configuration forms a relative equilibrium, then calculating the angular rotation of the configuration about the center-of-vorticity vector. We move in a rotating frame of reference so that the configuration is fixed, identify the separatrices in the flowfield, and plot the global streamline patterns as a function of the pole strengths and latitudinal positions of the rings. We carry the procedure out for single and double von Kármán vortex streets, with and without pole vortices. The single von Kármán street configurations are comprised of n evenly spaced vortices on each of two rings that symmetrically straddle the equator and are skewed with respect to each other by half a wavelength, while the double von Kármán ring configurations are made up of four rings of n evenly spaced vortices symmetrically straddling the equator.