In this paper we investigate the stability of a bilayer exposed to air flow. The bilayer consists of a viscoelastic solid layer (mucus), which rests on a viscous fluid film (serous fluid). The motivation behind this work is to examine the coupled, fluid/elastic instabilities related to mucus clearance in the lung where breathing and cough apply shear forces from the air flow onto the bilayer. Previous research on mucus transport due to air flow has not addressed the effects of the underlying serous layer nor those of surface tension at the mucus–air interface, two new features incorporated into the model. Surface tension effects are governed by the new parameter κ′ = (σ/dG′) where σ is the air–mucus surface tension, G′ is the elastic shear modulus of the mucus, and d is a characteristic thickness of the bilayer. The model predictions for the onset of unstable waves as a function of the parameters are compared to previous theories and experiments to provide physical interpretations and to compare results. The comparison with experiments show good qualitative and quantitative agreement. The results are compared, also, to flow over a single, viscoelastic layer, with no viscous fluid underneath, to demonstrate the appearance of new wave behaviour when the viscous fluid is added.