Context: The very young σ Orionis cluster (~3 Ma) is a cornerstone in understanding the formation of stars and substellar objects down to planetary masses. However, its stellar population is far from being completely known. Aims: This study's purpose is to identify and characterise the most massive stars of σ Orionis to complement current and future deep searches for brown dwarfs and planetary-mass objects in the cluster. Methods: I have cross-correlated the sources in the Tycho and 2MASS catalogues in a region of 30 arcmin radius with its centre in the O-type star σ Ori A. In this area, I studied the membership in the Ori OB 1b association of the brightest stars in the optical using astrometric, X-ray, and both infrared and optical photometric data from public catalogues, and spectroscopic data from the literature. Results: A list of 26 young stars, four candidate young stars, and 16 probable foreground stars has arisen from the study. Seven young stars probably harbour discs (four are new). There is no mass dependence of the disc frequency in the cluster. I have derived the first mass spectrum for σ Orionis from 1.1 to 24 M_&sun; (α = +2.0+0.2-0.1; roughly Salpeter-like). I also provide additional proof of the existence of several spatially superimposed stellar populations in the direction of σ Orionis. Finally, the cluster may be closer and older than previously thought.