  In the present paper, we define and study $C$-parallel and $C$-proper slant
curves of $S$-manifolds. We prove that a curve $\gamma $ in an $S$-manifold of
order $r\geq 3,$ under certain conditions, is $C$-parallel or $C$-parallel in
the normal bundle if and only if it is a non-Legendre slant helix or Legendre
helix, respectively. Moreover, under certain conditions, we show that $\gamma $
is $C$-proper or $C$-proper in the normal bundle if and only if it is a
non-Legendre slant curve or Legendre curve, respectively. We also give two
examples of such curves in $\mathbb{R}^{2m+s}(-3s).$
