I read from somewhere that astronomer use X-ray to observe the "shaking" not to be confused with wobbling of a distant neutron star, I think it is similar to what geologist using microphone to detecting sound wave (vibration) inside the earth (no pun) to find out its composition. How does a tightly packed super dense object like neutron star generate starquake? is this neutron star signature move... i mean trait?
I'm not sure what the current status of neutron starquake theory is, but last time I looked there were a couple of ideas.
The outer parts of a neutron star consist of a solid crust of neutron-rich nuclei (plus electrons, and a little deeper, free neutrons). Neutron stars are born rotating extremely fast, but they spin down due to the loss of rotational kinetic energy through magnetic dipole radiation. As a result, they are born oblate, but as the spin rate decreases this is not the lowest energy configuration. It used to be thought that starquakes (pulsar glitches) were caused by cracking of the crust and rearrangement towards a more spherical configuration.
The second idea is that it could be due to changes in the very intense magnetic field of the neutron star. The magnetic field is tied very tightly to the crustal material, so if the magnetic field shifts, possibly associated with the neutron star spin down, then it will put the crust under stress, which can be relieved by a starquake.
White dwarf stars also contain solid, crystalline material, however they do not spin so fast and do not have similar ultra-strong magnetic fields. I have not heard that they experience starquakes.
"Normal stars" do not have a crust or any other solid layer, so there can be no similar starquake phenomenon.