The morphological stability of interphase boundaries is a challenging problem, both from a mathematical and a physical point of view. In materials that are exposed to thermodynamic potential gradients, i.e., gradients of chemical potentials, electrical potential, temperature, or pressure, transport processes of the mobile components occur. These transport processes may cause morphological instabilities of the material surfaces and interfaces. In this paper, a comprehensive formal treatment of the coupled morphological stability of multiple phase boundaries will be given for oxides that are exposed to an oxygen potential gradient. In Part I, we study a single oxide layer, that is the morphological stability of two-boundaries; two oxide layers, $i.e.$ the case of three boundaries will be treated in the Part II. To explore the stability of diffusionally interacting boundaries an original, new method is applied. Based on an integral transformation of a special kind, this method reveals the evolution of the multiple boundaries’ perturbations without solving the diffusional problem inside the layer. The study of the morphological stability of the stationary moving boundaries is thereby reduced to the exploration of the singular points (in the complex plane) of the corresponding integrands.