Abstract The connectivity is an important measurement for the fault-tolerance of networks. To provide more accurate measures for the fault-tolerance of networks than the connectivity, some generalizations of connectivity have been introduced. Substructure connectivity and structure connectivity are two extended concepts of classical connectivity. As a variant of the popular network hypercube, the Möbius cubes is also a famous interconnection network in parallel and distributed systems. In this article, we calculate $H$-substructure connectivity and $H$-structure connectivity of Möbius cubes when $H$ is isomorphic to $P_{m},C_{m}$ and $K_{1,m}$.