We present a novel, deep-learning-based method—dubbed Galactic-Seismology Substructures and Streams Hunter, or GS3 Hunter for short—to search for substructures and streams in stellar kinematics data. GS3 Hunter relies on a combined application of Siamese neural networks to transform the phase space information and the K-means algorithm for the clustering. As a validation test, we apply GS3 Hunter to a subset of the Feedback in Realistic Environments (FIRE) cosmological simulations. The stellar streams and substructures thus identified are in good agreement with corresponding results reported earlier by the FIRE team. In the same vein, we apply our method to a subset of local halo stars from the Gaia Early Data Release 3 and GALAH DR3 data sets and recover several previously known dynamical groups, such as Thamnos 1+2, the hot thick disk, ED-1, L-RL3, Helmi 1+2, Gaia-Sausage-Enceladus, Sequoia, Virgo Radial Merger, Cronus, and Nereus. Finally, we apply our method without fine-tuning to a subset of K giant stars located in the inner halo region, obtained from the LAMOST Data Release 5 data set. We recover three previously known structures (Sagittarius, Hercules-Aquila Cloud, and the Virgo Overdensity), but we also discover a number of new substructures. We anticipate that GS3 Hunter will become a useful tool for the community dedicated to the search for stellar streams and structures in the Milky Way (MW) and the Local Group, thus helping advance our understanding of the stellar inner and outer halos and the assembly and tidal stripping history in and around the MW.