The advances of the theoretical modeling and detection with the superconducting gravimeters (SGs), related to the Slichter triplet, are systematically introduced in this paper. The translational oscillations of the Earth’s solid inner core, which is usually called as the Slichter modes in honor of the scientist who firstly pointed out its theoretical existence, are a triplet of the principle normal modes of the Earth. The triplet is primarily driven by the gravitational forces and has eigenperiods of several hours. From the theoretical modeling, it is found that the eigenperiods of the Slichter modes are most sensitive to the density contrast across the inner core boundary (ICB), and are only slightly disturbed by the viscosity of the fluid near the ICB, the transitional zone between the outer and inner cores, Lorentz forces and some others effects. Therefore, study and detection of the Slichter modes may provide significant information about the structure of the Earth’s deep interior. Due to less knowledge to the Earth’s deep interior, there exists great uncertainty in the results of theoretical modeling based on the present Earth’s models. It is necessary to detect the signatures related to the Slichter modes by using the ground-based observations. Smylie (1992) claimed that the weak gravity resonance signatures, coming from the Slichter triplet, were found in the SG data from plural stations in the central Europe. Based on the results, the density and viscosity near the ICB have been evaluated. However, thses results were questioned by the other scientists. Recently, we have comprehensively analyzed and treated the continuous long-term SG data from the worldwide network of the Global Geodynamics Projects. It is found that there is no evidence to support the existence of the resonance related to the Slichter triplet as what Smylie claimed. On the contrary, it is found that there are eight significant spectral peaks related to some globally geodynamical phenomena, and the Slichter triplet is one of their possible resources. 