  Given a constructible sheaf $F$ on a complex manifold, Kashiwara-Schapira
defined the notion of singular support and characteristic cycle of $F$. On the
other hand for a Zariski constructible \'{e}tale sheaf $F$ on an algebraic
variety $X$, Beilinson defined the notion of singular support of $F$ and Saito
defined the notion of characteristic cycle of $F$. In this article we compare
these notions and prove that they agree in a suitable sense. In the appendix we
discuss extension of the notions of singular support and characteristic cycles
developed by Umezaki-Yang-Zhao and Barrett.
