 The stability behavior of Rayleigh-Taylor and Landordarius instabilities in ablation fronts and laminar flames, respectively, is studied using a physical model that identifies the mechanisms controlling their stability based on the energy transport driving them. The evolution of perturbations due to instability always leads to changes in temperature gradients, but only if the driving mechanism is sensitive to these changes, such as thermal conduction in ablation fronts, does it result in restoring force. In flame fronts, the driving mechanism is not sensitive to temperature gradient perturbations, but instead to temperature perturbations, which induces instability even without a gravitational field. The forces driving flame instability and providing stabilization to ablation fronts are obtained from the same theoretical framework, and the stabilizing role of lateral thermal conduction for short perturbation wavelengths is analyzed. This article was authored by A. R. Pires and N. A. Tahir.