Electromagnetic waves are transverse in nature.What are some other kind of non electromagnetic waves which are also transverse in nature?
A violin string. Vibrating strings are typically the first time that physics students encounter waves. The (one-dimensional) wave equation emerges naturally from considering the acceleration of a small string segment due to its tension and curvature in the limit of small transverse displacement.
Transverse waves happen in elastic solids. The oscillations are made as the solid atoms/molecules are displaced away from their relaxed position, in directions perpendicular to the propagation of the wave. This sort of wave is also called shear waves, secondary waves or S-waves, esp. in seismology. The velocity of a shear wave is controlled by the shear modulus ($G$) as follows:
where $\rho$ is the density of the solid.
In addition to a string or a whip, transverse waves exist on a membrane of a drum, in elastic solids (shear waves, such as secondary or S-waves in seismology), waves on the surface of a liquid (e.g. a lake), waves on the surface of an inflated balloon (e.g. a soap bubble). Some of transverse waves can coexist with longitudinal waves. E.g., both types can propagate at the same time in elastic solids. Gravitational waves are also transverse.