Technically there are 4 types; P-wave(primary), S-wave(secondary), Love wave and Rayleigh wave,
the love and rayleigh are surface waves and they are difficult to differentiate on seismograph and so are counted as one. Although all three are equally damaging, by damage it means intensity of the earthquake. The intensity depends upon the infrastructure, for instance, mason structures are stable and strong it requires are really strong shock to be damaged.
When an earthquake takes place, four types of seismic waves travel from the epicenter.
The fastest traveling wave is the P-waves; Primary waves or P-waves AKA compressional waves that are longitudinal in nature. These waves can travel through any type of material, including fluids, and can travel at nearly twice the speed of S-waves. Typical speeds are 330 m/s in air, 1450 m/s in water and about 5000 m/s in granite.
Next is the S-waves, Secondary waves or S-waves AKA shear waves that are transverse in nature. They travel perpendicular to the direction of propagation. S-waves can travel ONLY through solids. S-waves are slower than P-waves, and speeds are typically around 60% of that of P-waves in any given material.
Surface waves travel along the , they diminish as they get further from the surface. Their velocity is lower than those of body waves (P and S). their speeds are slower to others.
Rayleigh waves travel as ripples. They are slower than body waves. In the layered medium (like the crust and upper mantle) the velocity of the Rayleigh waves depends on their frequency and wavelength.
Love waves are horizontally polarized shear waves or S-H waves They usually travel slightly faster than Rayleigh waves, about 90% of the S-wave velocity, and have the largest amplitude.