The Australia Telescope 20 GHz (AT20G) Survey is a blind survey of the whole Southern sky at 20 GHz (with follow-up observations at 4.8 and 8.6 GHz) carried out with the Australia Telescope Compact Array (ATCA) from 2004 to 2007. The Bright Source Sample (BSS) is a complete flux-limited subsample of the AT20G Survey catalogue comprising 320 extragalactic (|b|>1.5 deg) radio sources south of dec = -15 deg with S(20 GHz) > 0.50 Jy. Of these, 218 have near simultaneous observations at 8 and 5 GHz. In this paper we present an analysis of radio spectral properties in total intensity and polarisation, size, optical identifications and redshift distribution of the BSS sources. The analysis of the spectral behaviour shows spectral curvature in most sources with spectral steepening that increases at higher frequencies (the median spectral index \alpha, assuming S\propto \nu^\alpha, decreases from \alpha_{4.8}^{8.6}=0.11 between 4.8 and 8.6 GHz to \alpha_{8.6}^{20}=-0.16 between 8.6 and 20 GHz), even if the sample is dominated by flat spectra sources (85 per cent of the sample has \alpha_{8.6}^{20}>-0.5). The almost simultaneous spectra in total intensity and polarisation allowed us a comparison of the polarised and total intensity spectra: polarised fraction slightly increases with frequency, but the shapes of the spectra have little correlation. Optical identifications provided an estimation of redshift for 186 sources with a median value of 1.20 and 0.13 respectively for QSO and galaxies.