Context. Sulphur is a relatively abundant element in the local galaxy which is known to form a variety of molecules in the circumstellar envelopes of AGB stars. The abundances of these molecules vary based on the chemical types and mass-loss rates of AGB stars. Aims. Through a survey of (sub-)millimetre emission lines of various sulphur-bearing molecules, we aim to determine which molecules are the primary carriers of sulphur in different types of AGB stars. In this paper, the first in a series, we investigate the occurrence of H 2 S in AGB circumstellar envelopes and determine its abundance, where possible. Methods. We have surveyed 20 AGB stars with a range of mass-loss rates and of different chemical types using the APEX telescope to search for rotational transition lines of five key sulphur-bearing molecules: CS, SiS, SO, SO 2 and H 2 S. Here we present our results for H 2 S, including detections, non-detections and detailed radiative transfer modelling of the detected lines. We compare results based on different descriptions of the molecular excitation of H 2 S and different abundance distributions, including Gaussian abundances, where possible, and two different abundance distributions derived from chemical modelling results. Results. We detected H 2 S towards five AGB stars, all of which have high mass-loss rates ofṀ ≥ 5 × 10 −6 M yr −1 and are oxygenrich. H 2 S was not detected towards the carbon or S-type stars that fall in a similar mass-loss range. For the stars in our sample with detections, we find peak o-H 2 S abundances relative to H 2 between 4 × 10 −7 and 2.5 × 10 −5 . Conclusions. Overall, we conclude that H 2 S can play a significant role in oxygen-rich AGB stars with higher mass-loss rates, but is unlikely to play a key role in stars of other chemical types or the lower mass-loss rate oxygen-rich stars. For two sources, V1300 Aql and GX Mon, H 2 S is most likely the dominant sulphur-bearing molecule in the circumstellar envelope.