A mathematical analysis has been performed to investigate the combined effects of viscosity and elasticity on the interfacial and the surface stabilities of two superposed Oldroyd-B fluids flowing down an inclined plane. Our results indicate that arranging the less viscous and more elastic liquid as the skin layer is helpful in avoiding the occurrence of interfacial waves. For the stability of free surface, it is found that the elasticity in each layer is always destabilizing free surface waves. However, this destabilization can be lessened by designing the more viscous liquid as the skin layer. In addition, the effect of increasing Reynolds number on the interface and the free surface is investigated. It is shown that increasing Reynolds number can stabilize or destabilize the interface, depending on the parametric space of the flow system. However, free surface is always destabilized by an increase in Reynolds number. In general, our analysis indicates that under certain physical conditions (i.e., viscosity and elasticity ratios) and geometric conditions (i.e., depth ratio), stable operations in the coating process can be attained.