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.
https://socratic.org/questions/which-of-the-three-types-of-seismic-waves-causes-the-most-damage-during-an-earth
With all due respect to the jumper, the landing is a good metaphor for an earthquake. When the rocks along either side of a fault shift past each other, they send out energy in waves, much like the cannon-baller's sploosh rippled out. And like those waves that go through the pool, the waves of energy caused by the rupture pass through the earth and we feel them as an earthquake.
http://www.exploratorium.edu/faultline/basics/waves.html
P and S waves may seem like shaky business, but the waves that really do damage are the ones that occur when the energy of the quake reaches the surface of the earth. Rayleigh waves churn over and under like rolling ocean waves; Love waves shake the earth from side to side.
http://www.exploratorium.edu/faultline/basics/waves.html
S-waves travel more slowly, usually at 2.5–4 km/sec (9000–14,000 km/h). Sound waves are usually called P-waves and are heard but not often felt. Except in the most powerful earthquakes they generally do not cause much damage.
http://www.sms-tsunami-warning.com/pages/seismic-waves
What seismic wave causes the most damage? A: During an earthquake, surface waves are the seismic waves that cause the most damage. Unlike other seismic waves that move deep inside the Earth, surface waves move along just under the surface of the Earth like waves in water. Continue Reading.
https://www.reference.com/science/seismic-wave-causes-damage-7b01b0eb10fa3e41
An earthquake also causes secondary or shear waves, called S waves. These travel at about half the speed of P waves, but can be much more destructive. S waves move the earth perpendicularly to the direction the wave is traveling. Picking up the Slinky again, move it left and right, rather than forward and back.
http://www.exploratorium.edu/faultline/basics/waves.html
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.
https://socratic.org/questions/which-of-the-three-types-of-seismic-waves-causes-the-most-damage-during-an-earth
Surface waves are the slowest seismic waves and are the final waves to hit an area after an earthquake. First, P-waves move through compression through the deep layers of the Earth. P-waves are also carried by air and water, causing them to make a sound when they hit the Earth's surface. They result in minimal damage.
https://www.reference.com/science/seismic-wave-causes-damage-7b01b0eb10fa3e41
Body waves are of two types: compressional or primary (P) waves and shear or secondary (S) waves. P- and S- waves are called body waves because they can travel through the interior of a body such as the Earth's inner layers, from the focus of an earthquake to distant points on the surface.
http://www.sms-tsunami-warning.com/pages/seismic-waves
Surface waves, in contrast to body waves can only move along the surface. They arrive after the main P and S waves and are confined to the outer layers of the Earth. They cause the most surface destruction. Earthquake surface waves are divided into two different categories: Love and Rayleigh. Love waves have a particle motion, which, like the S-wave, is transverse to the direction of propagation but with no vertical motion. Their side-to-side motion (like a snake wriggling) causes the ground to twist from side to side, that’s why Love waves cause the most damage to structures.
http://www.sms-tsunami-warning.com/pages/seismic-waves
what types of earthquake waves usually cause the most destruction?
The primary and secondary types of earthquake waves usually cause the most destruction. The P and S type of waves in an earthquake that usually cause the most destruction.