  Recently, the singular support and the characteristic cycle of an \'etale
sheaf on a smooth variety over a perfect field are constructed by Beilinson and
Saito, respectively. In this article, we extend the singular support to a
relative situation. As an application, we prove the generic constancy for
singular supports and characteristic cycles of \'etale sheaves on a smooth
fibration. Meanwhile, we show the failure of the lower semi-continuity of
characteristic cycles in a higher relative dimension case, which is different
from Deligne and Laumon's result in the relative curve case.
