  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 sigma
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 sigma
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, H-alpha, 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 H-alpha of 34 and 12 stars, respectively. We
identified 11 close (rho <~ 3 arcsec) binaries resolved by Gaia, of which three
are new, 14 strong accretors, of which four are new and another four have
H-alpha emission shifted by over 120 km/s, 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 sigma Orionis members and taken into account in
high-impact works on, for example, disc frequency and initial mass function.
