There are 4 US Post Offices in Fleming County serving a total of 14,785 residents equating to 3,696 residents per post office. It's estimated that approximately 23,599 packages are handled across the 4 county post offices annually. Fleming County is a county located in the U.S. state of Kentucky. As of the 2010 census, the population was 14,348. Its county seat is Flemingsburg. The county was formed in 1798 and named for Colonel John Fleming, an Indian fighter and early settler. It's a prohibition or dry county. In 1998, the Kentucky General Assembly designated Fleming County as the Covered Bridge Capital of Kentucky."