Context. Sulphur is one of the most abundant elements in the Universe. Surprisingly, sulphuretted molecules are not as abundant as expected in the interstellar medium and the identity of the main sulphur reservoir is still an open question. Aims. Our goal is to investigate the H2S chemistry in dark clouds, as this stable molecule is a potential sulphur reservoir. Methods. Using millimeter observations of CS, SO, H2S, and their isotopologues, we determine the physical conditions and H2S abundances along the cores TMC 1-C, TMC 1-CP, and Barnard 1b. The gas-grain model NAUTILUS is used to model the sulphur chemistry and explore the impact of photo-desorption and chemical desorption on the H2S abundance. Results. Our modeling shows that chemical desorption is the main source of gas-phase H2S in dark cores. The measured H2S abundance can only be fitted if we assume that the chemical desorption rate decreases by more than a factor of 10 when nH > 2 × 104. This change in the desorption rate is consistent with the formation of thick H2O and CO ice mantles on grain surfaces. The observed SO and H2S abundances are in good agreement with our predictions adopting an undepleted value of the sulphur abundance. However, the CS abundance is overestimated by a factor of 5-10. Along the three cores, atomic S is predicted to be the main sulphur reservoir. Conclusions. The gaseous H2S abundance is well reproduced, assuming undepleted sulphur abundance and chemical desorption as the main source of H2S. The behavior of the observed H2S abundance suggests a changing desorption efficiency, which would probe the snowline in these cold cores. Our model, however, highly overestimates the observed gas-phase CS abundance. Given the uncertainty in the sulphur chemistry, we can only conclude that our data are consistent with a cosmic elemental S abundance with an uncertainty of a factor of 10.