Usually in the study of singularity problems, only the most critical singular order is considered. For three-dimensional interface corner problems, if only the most critical singular order of stresses is considered, it is possible to lose the opportunity to compute the full modes of stress intensity factors. To fully understand the failure behavior of three-dimensional interface corners, a definition of the stress intensity factors for the lower singular orders is proposed in this paper based on that of the most critical singular order. Moreover, to compute the proposed multi-order stress intensity factors accurately and efficiently, a path-independent $H$-integral, which has been proven useful for the two-dimensional interface corners, is now modified into a domain-independent $H$-integral for the three-dimensional interface corner problems. Because the stress intensity factors characterize the fracture behavior focused on an arbitrary tip along the corner front, based on anisotropic elasticity the near tip solutions and complementary solutions of two-dimensional generalized plane strain problems are introduced and then utilized for computation of three-dimensional $H$-integral. To illustrate the validity of the present work, several three-dimensional numerical examples are analyzed and compared with the existing published solutions. Finally, two examples about the interface corners, which occur frequently in electric packages, are solved to show the feasibility and practicability of the proposed approach.