Interfaces between stratified epithelia and their supporting stromas commonly exhibit irregular shapes. Undulations are particularly pronounced in dysplastic tissues and typically evolve into long, finger-like protrusions in carcinomas. In previous work (Basan et al 2011 Phys. Rev. Lett. 106 158101), we demonstrated that an instability arising from viscous shear stresses caused by the constant flow due to cell turnover in the epithelium could drive this phenomenon. While interfacial tension between the two tissues as well as mechanical resistance of the stroma tend to maintain a flat interface, an instability occurs for sufficiently large viscosity, cell-division rate and thickness of the dividing region in the epithelium. Here, extensions of this work are presented, where cell division in the epithelium is coupled to the local concentration of nutrients or growth factors diffusing from the stroma. This enhances the instability by a mechanism similar to that of the Mullins–Sekerka instability in single-diffusion processes of crystal growth. We furthermore present the instability for the generalized case of a viscoelastic stroma.