Through the whole medium, mechanical transverse waves cannot propagate.
Solids can only propagate mechanical transverse waves if they have a high shear modulus of elasticity. Because there is no mechanism for propelling motion perpendicular to the wave's propagation, they can only move through solids and not in a gas or a liquid. Although longitudinal waves may travel through any sort of material, they cannot travel through a vacuum since a medium is required for their travel. So answer is mechanical transverse waves can propagate through solids only.