  In this article, we introduce the concepts of graded $s$-prime submodules
which is a generalization of graded prime submodules. We study the behavior of
this notion with respect to graded homomorphisms, localization of graded
modules, direct product, and idealization. We succeeded to prove the existence
of graded $s$-prime submodules in the case of graded-Noetherian modules. Also,
we provide some sufficient conditions for the existence of such objects in the
general case, as well as, in the particular case of grading by $\mathbb{Z}$, a
finite group, or a polycyclic-by-finite group, in addition to crossed product
grading.
