The stability of stationary spots on cathodes of arcs in vacuum and ambient gas is investigated by means of the simulation of the temporal evolution of perturbations imposed over steady-state solutions. Two cases of loading conditions are considered, namely, spots operating at a fixed current (the case typical of small-scale experiments) and spots operating at a fixed voltage (the case typical of high-power circuit breakers). Results are reported on spots on large copper cathodes of vacuum arcs and on spots on tungsten cathodes of high-pressure argon arcs. It is shown, in particular, that if the ballast resistance in small-scale laboratory experiments with a high-current arc is insufficient, the potential consequence may be a thermal explosion of a spot, if the arc burns in vacuum, and massive melting of the cathode surface, if the arc burns in ambient gas. This conclusion conforms to trends observed in the experiment.