The radio spectral index is a powerful probe for classifying cosmic radio sources and understanding the origin of the radio emission. Combining data at 147MHz and 1.4GHz from the TIFR GMRT Sky Survey (TGSS) and the NRAO VLA Sky Survey (NVSS), we produced a large-area radio spectral index map of ~80 per cent of the sky (Dec. >-40deg), as well as a radio spectral index catalogue containing 1396515 sources, of which 503647 are not upper or lower limits. Almost every TGSS source has a detected counterpart, while this is true only for 36 per cent of NVSS sources. We released both the map and the catalogue to the astronomical community. The catalogue is analysed to discover systematic behaviours in the cosmic radio population. We find a differential spectral behaviour between faint and bright sources as well as between compact and extended sources. These trends are explained in terms of radio galaxy evolution. We also confirm earlier reports of an excess of steep-spectrum sources along the galactic plane. This corresponds to 86 compact and steep-spectrum source in excess compared to expectations. The properties of this excess are consistent with normal non-recycled pulsars, which may have been missed by pulsation searches due to larger than average scattering along the line of sight.