Isotropic solid materials can support the propagation of both dilatational waves and distortional waves, the latter better known as shear waves. While propagation and scattering of elastic waves has an established history within the geophysics community, interest in elastic wave propagation in soft biological materials is relatively recent. Nevertheless, the medical imaging context presents a number of exotic situations to study shear wave phenomena, including in the presence of strong magnetic fields, superposed large poroelastic deformations, strong material property gradients, and medium activation. It also presents novel inverse problem opportunities. We use mathematical models to study elastic shear wave propagation in these scenarios.