Question

In solids both longitudinal and transverse waves are possible,
but transverse waves are not produced in gases. Why?

Answer 1

In solids, the shear strain creates a shear stress and therefore creates internal forces in the solid. In fluids (liquids or gases) the shear stress is zero (zero shear stress is one way to define what you mean by "a fluid"), so there are no internal forces, therefore a shear wave cannot travel through the fluid. 
A longitudinal wave creates axial strain which creates axial stress in solids, or pressure in fluids. Pressure does create internal forces in fluids, so longitudinal waves are possible.