There are 12 US Post Offices in Johnson County serving a total of 20,009 residents equating to 1,667 residents per post office. It's estimated that approximately 31,937 packages are handled across the 12 county post offices annually. Johnson County is a county located in the U.S. state of Kentucky. As of the 2010 census, the population was 23,356. Its county seat is Paintsville. The county was formed in 1843 and named for Richard Mentor Johnson, War of 1812 general, United States Representative, Senator, and Vice President of the United States. Johnson County is classified as a moist county, which is a county in which alcohol sales are not allowed (a dry county), but containing a \"wet\" city, in this case Paintsville, where alcoholic beverage sales are allowed."