Which of the following statements are true for wave motion? With propagation of longitudinal waves...

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.