We present an analysis of the stability of a dendrite against a small perturbation to the tip velocity. We find that dendritic growth in pure metals and alloys will become unstable above some upper critical undercooling ${\mathrm{\Delta T}}_2^{*}.$ In alloys above a critical concentration, dendritic growth may also become unstable below a lower critical undercooling, ${\mathrm{\Delta T}}_1^{*}.$ In the example systems studied, Ni–Cu and Ag–O, the location of these unstable regions shows remarkably close agreement with the onset of spontaneous grain refinement. We obtain values of ${\mathrm{\Delta T}}_2^{*}$ for Ni and Ag of 195 K and 160 K, respectively, in good agreement with the observed values of 170 K and 133–153 K.