Shales are the most common sedimentary rocks in hydrocarbon environments often forming the source rock and trapping rock for a reservoir. Due to the platey nature of the constituent grains, shales are commonly anisotropic. In this paper we calculate seismic waveforms for highly anisotropic shales using Maslov asymptotic theory (MAT). This theory is an extension of classical ray theory which provides valid waveforms in regions of caustics (wavefront folding) where ray theory amplitudes are unstable. Asymptotic ray theory (ART) is based on the Fermat or geometrical ray which connects the source and receiver. In contrast, the Maslov solution integrates the contributions from neighbouring non-Fermat rays. Raypaths, travel-times, amplitudes and synthetic seismograms are presented for three highly anisotropic shares using a very simple Ib model comprised of an anisotropic share overlying an isotropic shale. The ART waveforms fail to account for complex waveform effects due to triplications. In comparison, the MAT waveforms predict nonsingular amplitudes at wavefront cusps and it predicts the diffracted signals from these cusps. A Maslov solution which integrates ray contributions over a single slowness component will break down when rays focus in 3D (at a point rather than along a line). One of the tested shares shows such a point caustic and integration over 2 slowness components is required to remove the amplitude singularity. Finally, we examine the effects of wavefront triplications on Alford rotations which are used to estimate shearwave splitting. In such cases, the rotation successfully finds the fast shear-wave polarization, but it can be unreliable in its estimate of the time separation.