We investigate a new mechanism for instability (named divergent instability), characterized by the formation of azimuthal cells, and find it to be a generic feature of three-dimensional steady axisymmetric flows of viscous incompressible fluid with radially diverging streamlines near a planar or conical surface. Four such flows are considered here: (i) Squire-Wang flow in a half-space driven by surface stresses; (ii) recirculation of fluid inside a conical meniscus; (iii) two-cell regime of free convection above a rigid cone; and (iv) Marangoni convection in a half-space induced by a point source of heat (or surfactant) placed at the liquid surface. For all these cases, bifurcation of the secondary steady solutions occurs: for each azimuthal wavenumber m = 2, 3,…, a critical Reynolds number (Re*) exists. The intent to compare with experiments led us to investigate case (iv) in more detail. The results show a non-trivial influence of the Prandtl number (Pr): instability does not occur in the range 0.05 < Pr < 1; however, outside this range, Re*(m) exists and has bounded limits as Pr tends to either zero or infinity. A nonlinear analysis shows that the primary bifurcations are supercritical and produce new stable regimes. We find that the neutral curves intersect and subcritical secondary bifurcation takes place; these suggest the presence of complex unsteady dynamics in some ranges of Re and Pr. These features agree with the experimental data of Pshenichnikov & Yatsenko (Pr = 103).