At elevated temperatures, stresses enhance mass transport at the microstructural scale and under certain conditions, lead to microstructural instabilities. Using a linear stability analysis, we examine the effect of stresses on the morphological stability of plate-like phases and lamellar eutectic composites. These stresses can be either due to misfit strains and/or due to externally applied loads. We find that for misfitting plates, the nominally flat plate-matrix interface is unstable with respect to the growth of perturbations with wavelengths greater than a critical wavelength, provided that the reinforcing plates are elastically suffer than the surrounding matrix. On the other hand, for stresses generated by externally applied loads, the flat interface is always unstable as long as the plate modulus and the matrix modulus are not identical. In addition, the analysis reveals that misfit strains can either counteract or enhance the destabilizing influence of applied loads depending on the elastic properties of the plate and the matrix. Finally, we have developed stability diagrams that identify material properties and operating conditions required to maintain a stable interface in these lamellar eutectic composites.