Context. In order to understand the evolution and feedback of active galactic nuclei (AGN) and star formation, it is important to use molecular lines as probes of physical conditions and chemistry. Aims. We use H2S to investigate the impact of starburst and AGN activity on the chemistry of the molecular interstellar medium in luminous infrared galaxies. Specifically, our aim is to search for evidence of shock enhancement of H2S related to galactic-scale mechanical feedback processes such as outflows. Methods. Using the APEX single-dish telescope, we have observed the 1(10)-1(01) transition of ortho-H2S at 168 GHz towards the centres of 12 nearby luminous infrared galaxies. We have also observed the same line towards the ultra-luminous infrared galaxy Mrk 231 with the NOEMA interferometer. Results. We detected H2S towards NGC 253, NGC 1068, NGC 3256, NGC 4418, NGC 4826, NGC 4945, Circinus, M 83, and Mrk 231. Upper limits were obtained for NGC 1097, NGC 1377, and IC 860. We also detected line emission from HCN 2-1 in all galaxies in the APEX survey as well as HCO+, HNC, CH3CN, CH3OH, H2CS, HOC+, and SO in several of the sample galaxies. Mrk 231 has a rich 2 mm molecular spectrum and, in addition to H2S, we detect emission from HC3N, CH3OH, (HCO+)-O-18, C2S, and CH3CCH. Four galaxies show elevated H2S emission relative to HCN: Circinus, NGC 3256, NGC 4826, and NGC 4418. We suggest that the high line ratios are caused by elevated H2S abundances in the dense gas. However, we do not find any clear connection between the H2S/HCN line intensity ratio and the presence (or speed) of molecular outflows in the sample galaxies. Therefore, H2S abundances do not seem to be globally affected by the large-scale outflows. In addition, the H2S/HCN line ratio is not enhanced in the line wings compared to the line core in Mrk 231. This suggests that H2S abundances do not increase in the dense gas in the outflow. However, we do find that the H2S and HCN luminosities (L-H2S and L-HCN) correlate well with the total molecular gas mass in the outflow, M-outflow(H-2), in contrast to L-CO and LHCO+. We also find that the line luminosity of H2S correlates with the total infrared luminosity in a similar way as that of H2O. Conclusions. We do not find any evidence of H2S abundance enhancements in the dense gas due to galactic-scale outflows in our sample galaxies, nor in the high-resolution study of Mrk 231. We discuss possible mechanisms behind the suggested H2S abundance enhancements in NGC 4418, Circinus, NGC 3256, and NGC 4826. These include radiative processes (for example X-rays or cosmic rays) or smaller-scale shocks. Further high-resolution and multi-transition studies are required to determine the cause behind the elevated H2S emission in these galaxies. We suggest that L-H2S serves as a tracer of the dense gas content, similar to L-HCN, and that the correlation between L-H2S and M-outflow(H-2) implies a relation between the dense gas reservoir and the properties and evolution of the molecular feedback. This potential link requires further study since it holds important keys to our understanding of how the properties of molecular outflows relate to those of their host galaxies. Finally, the similar infrared-correlation coefficients between H2S and H2O may indicate that they originate in the same regions in the galaxy: warm gas in shocks or irradiated by star formation or an AGN.
