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SH$^+$ is a surprisingly widespread molecular ion in diffuse interstellar clouds. There, it plays an important role triggering the sulfur chemistry. In addition, SH$^+$ emission lines have been detected at the UV-illuminated edges of dense molecular clouds, \mbox{so-called} photo-dissociation regions (PDRs), and toward high-mass protostars. An accurate determination of the SH$^+$ abundance and of the physical conditions prevailing in these energetic environments relies on knowing the rate coefficients of inelastic collisions between SH$^+$ molecules and hydrogen atoms, hydrogen molecules, and electrons. In this paper, we derive SH$^+$--H fine and hyperfine-resolved rate coefficients from the recent quantum calculations for the SH$^+$--H collisions, including inelastic, exchange and reactive processes. The method used is based on the infinite order sudden approach. State-to-state rate coefficients between the first 31 fine levels and 61 hyperfine levels of SH$^+$ were obtained for temperatures ranging from 10 to 1000 K. Fine structure-resolved rate coefficients present a strong propensity rule in favour of $Δj = ΔN$ transitions. The $Δj = ΔF$ propensity rule is observed for the hyperfine transitions. {The new rate coefficients will help significantly in the interpretation of SH$^+$ spectra from PDRs and UV-irradiated shocks where the abundance of hydrogen atoms with respect to hydrogen molecules can be significant.