What factors does the speed of a transverse wave depend on?

Transverse waves can travel through solids or any medium that can withstand shearing strain. Fluids, including liquids and gases, cannot sustain shear force, so such types of mechanical waves cannot travel through them. 

Longitudinal waves can travel through all elastic media that can sustain compressive strain. That means, these mechanical waves can travel through solids, liquids, and gases.  

Sound waves and water waves are mechanical waves, which always require a medium to travel. Light, however, does not require a medium to propagate, as it is electromagnetic.  Light can travel through vacuum, while sound and water waves cannot travel through vacuum.