Melbourne is a city and county seat of Izard County, Arkansas, United States. The population was 1,848 at the 2010 census. It is home to th…Melbourne is a city and county seat of Izard County, Arkansas, United States. The population was 1,848 at the 2010 census. It is home to the main campus of Ozarka College. Melbourne is located at 36°03′35″N 91°53′40″W / 36.059764°N 91.894396°W.