Neither Vishnu nor Brahma stay on the earth but why did lord Shiva choose Mount Kailash ( which is on earth) as his abode?
See the following passage from the Devi Bhagavata Purana (Book 3; Chapter 13):
- The two sons Priyavrata and Uttânapâda were born of Manu in the womb of S’atarûpâ and the three daughters, very beautiful and fair complexioned, were also born of him.
- Creating then, the Bhagavân, the lotus born Brahmâ built the beautiful Brahma-loka, on the top of the Meru mountain.
- Then the Bhagavân Visnu built the Vaikuntha city on the top of all the lokas or worlds to dwell with his consort Laksmî. 19. Mahâdeva, too, built the exceedingly beautiful Kailâs’a and stayed there with his Bhûtas and played with them at his will.
- The third Loka termed Heaven was built on the top of Meru, decorated with various precious gems and jewels and stones. It was fixed as the abode of Indra. 21-22. When the great ocean was churned, Pârijâta, the best of all the trees, the elephant Airâvata with four tusks, the Heavenly cow yielding the milk of all desires, the Uchchais’ravâ horse and Apsarâs, Rambhâ and others, arose and were taken by Indra. These became the ornaments of Heaven.
It is describing how creation happened one after another at the beginning. So, as per this passage, Lord Shiva "created" Kailasha as his abode and "not chose". Because it's his will.
One other explanation on "why he chose" is given in this answer though.