One of the smallest towns in Kentucky is New Castle, 35 minutes
away from Louisville, with a population of 919.
A town called Plymouth Village has a population of 201.
There are many, many towns in Kentucky with a population less
than 1000. (of course, it also depends on what you call a
"town").
Concord, Ky has a population of 28 (It's in the Northeast part
of the state, along the Ohio river).