Exact, non-planar travelling solutions of the eikonal equation on an infinite plane are presented for the first time. These solutions are matched to produce corrugated wave fronts and patterns such as 'spot' solutions as well as extended parabolic type wave fronts. The stability of these solutions is also analysed. The variational equation which belongs to a generalised Wangerin class of differential equations is solved, first with the aid of the Liouville-Green approximation for the estimated eigenvalues characterising stability and then by a more elaborate shooting-matching method. All of the three types of travelling solutions are found to be geometrically stable. It is suggested that some of these predictions are experimentally testable.