There are several websites that list Nashville, Tennessee, ZIP codes. City-Data.com, Zipmap.net and Hometownlocator.com all have ZIP code overlays on maps of the greater Nashville area. They also have zoom functions that allow for greater focus on a particular ZIP code.
Nashville's city government website, Nashville.gov, has a downloadable document showing a map with ZIP codes in relation to city council districts and school districts as well as other types of maps. However, if you want to have a tangible copy of a ZIP code map, local map stores are the way to go. The Nashville Yellow Pages has a list of those stores.