  We show that on any translation surface, if a regular point is contained in a
simple closed geodesic, then it is contained in infinitely many simple closed
geodesics, whose directions are dense in the unit circle. Moreover, the set of
points that are not contained in any simple closed geodesic is finite. We also
construct explicit examples showing that such points exist. For a surface in
any hyperelliptic component, we show that this finite exceptional set is
actually empty. The proofs of our results use Apisa's classifications of
periodic points and of $\GL(2,\R)$ orbit closures in hyperelliptic components,
as well as a recent result of Eskin-Filip-Wright.
