Summary: The work in this paper is divided into two parts. In the first part, we classify weakly prime \(\Delta \)-supergraded submodules and we compare them with prime \(\Delta \)-supergraded submodules. In the second part, we characterize primary \(\Delta \)-supergraded submodules of multiplication \(\Delta \)- supergraded modules, we study finitely generated multiplication \(\Delta \)- supergraded modules and then we give sufficient conditions for multiplication \(\Delta \)-supergraded modules to be cyclic.