We analyze travelling wave (TW) solutions for nonlinear systems consisting of an ODE coupled to a degenerate PDE with a diffusion coefficient that vanishes as the solution tends to zero and blows up as it approaches its maximum value. Stable TW solutions for such systems have previously been observed numerically as well as in biological experiments on the growth of cellulolytic biofilms. In this work, we provide an analytical justification for these observations and prove existence and stability results for TW solutions of such models. Using the TW ansatz and a first integral, the system is reduced to an autonomous dynamical system with two unknowns. Analysing the system in the corresponding phase-plane, the existence of a unique TW is shown, which possesses a sharp front and a diffusive tail, and is moving with a constant speed. The linear stability of the TW in two space dimensions is proven under suitable assumptions on the initial data. Finally, numerical simulations are presented that affirm the theoretical predictions on the existence, stability, and parametric dependence of the travelling waves. K. Mitra and S. Sonner would like to thank the Nederlandse Organisatie voor Wetenschappelijk Onderzoek (NWO) for their support through the Grant OCENW.KLEIN.358. K. Mitra is further supported by Fonds voor Wetenschappelijk Onderzoek (FWO) through the Junior Postodoctoral Fellowship. J. Hughes is supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada. H. Eberl acknowledges the financial support received from the Natural Sciences and Engineering Research Council of Canada (NSERC) through a Discovery Grant (RGPIN-2019-05003) and a Research Tools and Infrastructure Grant (RTI-2019-00317). We also thank C.J. van Duijn for fruitful discussions on stability, and the referees for their careful reading of the manuscript and valuable comments.