The dusty winds of cool evolved stars are a major contributor of the newly synthesised material enriching the Galaxy and future generations of stars. However, the details of the physics and chemistry behind dust formation and wind launching have yet to be pinpointed. Recent spatially resolved observations show the importance of gaining a more comprehensive view of the circumstellar chemistry, but a comparative study of the intricate interplay between chemistry and physics is still difficult because observational details such as frequencies and angular resolutions are rarely comparable. Aiming to overcome these deficiencies, ATOMIUM is an ALMA Large Programme to study the physics and chemistry of the circumstellar envelopes of a diverse set of oxygen-rich evolved stars under homogeneous observing conditions at three angular resolutions between ~0.02"-1.4". Here we summarize the molecular inventory of these sources, and the correlations between stellar parameters and molecular content. Seventeen oxygen-rich or S-type asymptotic giant branch (AGB) and red supergiant (RSG) stars have been observed in several tunings with ALMA Band 6, targeting a range of molecules to probe the circumstellar envelope and especially the chemistry of dust formation close to the star. We systematically assigned the molecular carriers of the spectral lines and measured their spectroscopic parameters and the angular extent of the emission of each line from integrated intensity maps. Across the ATOMIUM sample, we detect 291 transitions of 24 different molecules and their isotopologues. This includes several first detections in oxygen-rich AGB/RSG stars: PO v=1, SO2 v1=1 and v2=2, and several high energy H2O transitions. We also find several first detections in S-type AGB stars: vibrationally excited HCN 32 = 2, 3 and SiS 3 = 4, 5, 6, as well as first detections of the molecules SiC, AlCl, and AlF in W Aql. Overall, we find strong correlations between the following molecular pairs: CS and SiS, CS and AlF, NaCl and KCl, AlO and SO, SO2 and SO, and SO2 and H2O; meaning both molecules tend to have more detected emission lines in the same sources. The measured isotopic ratios of Si and S are found to be consistent with previous measurements, except for an anomalously high 29Si/30Si ratio of 4+/-1 in the RSG VX Sgr. This paper presents the overall molecular inventory and an initial analysis of the large ATOMIUM dataset, laying the groundwork for future work deriving molecular abundances and abundance profiles using radiative transfer modeling which will provide more rigorous tests for chemical models.