Many online family trees show this Solomon as a son of Thomas R. Staton (b. ca 1734) and Ann Moore (b. ca 1737), but that doesn’t seem to fit to me, since Ann would have been about 55 when he was born. It seems more likely that he is their grandson. It is possible that he is a son of Thomas by a later wife, though.

1820 census of Bath Co., Ky, Owingsville precinct, lists:
Solomon Slatin, 1 m 26-44.

1830 census of Bath Co., Kentucky, p. 219, line 5, lists:
Solomon Staton, 1 m < 5, 2 m 5-9, 1 m 30-39, 1 f < 5, 1 f 20-29.
(On the same page, the family of Samuel Staton who married Drusilla Sadler is at line 1, and the family of William Fielder and Elizabeth Sharp is at line 6.)

1840 census of Bath Co., Ky, East of Slate Creek, lists”
Sollomon Slayton, 1 m 40-49, 1 f 30-39; 2 m 15-19, 1 m 10-14; 1 f 10-14, 1 f 5-9.

1850 census of Bath Co., Kentucky, enumerated 9 Aug, lists on p. 56 as family #382:
Soloman Staton, 58, farmer, b. Va; Malinda, 47, b. Ky; Hannah, 21, b. Ky; Mary A., 17, b. Ky; George, 12, b. Ky.

The following household is their son Benjamin.

1860 census of Bath Co., Ky, Owingsville P.O., enumerated 7 Aug, lists on p. 199 as family #1436:
Solomon Staton, 68, farmer; Malinda, 47; Mary A., 24. All listed as born in Ky, although that is in conflict with other records.

Unless Solomon married two different wives both named Malinda, the age for Malinda is off by quite a bit, since it would make her only 7 or 8 years old in Feb 1821.

1870 census of Bath Co., Ky, Mudlick precinct, enumerated 12 Aug, lists on p. 22 as family #160/162:
Staton, Solomon, 70; Malinda, 72; Louisiana, 9. All born in Ky.

Louisiana is the daughter of their daughter Mary Ann, whom I have not found in 1870.