Motivated by the non-Newtonian properties of mucus and the bilayer nature of fluid lining in the pulmonary airways, we investigate surfactant transport on both single and bilayer fluid systems; the aim is twofold. First, we explore the influence of two principle rheological properties of mucus, yield stress and shear thinning, on the surfactant spreading behaviour. Secondly, in these airways, mucus, which has substantial non-Newtonian properties, overlies the periciliary liquid layer (PCL) which is primarily Newtonian, and we incorporate this bilayer structure into the analysis. This consists of the derivation of coupled spatio-temporal evolution equations describing the layer thicknesses and surfactant concentration. Subsequent analytical methods examine limiting cases where similarity variables can be usefully employed, and more generally numerical simulations are performed.