Context. Only a few open clusters are as important for the study of stellar and substellar objects, and their formation and evolution, as the young σ Orionis cluster. However, a complete spectroscopic characterisation of its whole stellar population is still missing. Aims. We filled most of that gap with a large spectroscopic and astrometric survey of targets towards σ Orionis. Eventually, it will be one of the open clusters with the lowest proportion of interlopers and the largest proportion of confirmed cluster members with known uncontrovertible youth features. Methods. We acquired 317 low-resolution optical spectra with the Intermediate Dispersion Spectrograph (IDS) at the 2.5 m Isaac Newton Telescope (INT) and the Optical System for Imaging and low Resolution Integrated Spectroscopy (OSIRIS) at the 10.4 m Gran Telescopio Canarias (GTC). We measured equivalent widths of Li I, Hα, and other key lines from these spectra, and determined spectral types. We complemented this information with Gaia DR2 astrometric data and other features of youth (mid-infrared excess, X-ray emission) compiled with Virtual Observatory tools and from the literature. Results. Of the 168 observed targets, we determined for the first time spectral types of 39 stars and equivalent widths of Li I and Hα of 34 and 12 stars, respectively. We identified 11 close (ρ ≲ 3 arcsec) binaries resolved by Gaia, of which three are new, 14 strong accretors, of which four are new and another four have Hα emission shifted by over 120 km s⁻¹, two juvenile star candidates in the sparse population of the Ori OB1b association, and one spectroscopic binary candidate. Remarkably, we found 51 non-cluster-members, 35 of which were previously considered as σ Orionis members and taken into account in high-impact works on, for example, disc frequency and initial mass function.