A sample of 2712 radio-luminous galaxies is defined from the second data release of the Sloan Digital Sky Survey (SDSS) by cross-comparing the main spectroscopic galaxy sample with two radio surveys: the National Radio Astronomy Observatories (NRAO) Very Large Array (VLA) Sky Survey (NVSS, <VIII/65>) and the Faint Images of the Radio Sky at Twenty centimeters (FIRST, <VIII/71>) survey. The comparison is carried out in a multistage process and makes optimal use of both radio surveys by exploiting the sensitivity of the NVSS to extended and multicomponent radio sources in addition to the high angular resolution of the FIRST images. A radio source sample with 95 per cent completeness and 98.9% reliability is achieved, far better than would be possible for this sample if only one of the surveys was used. The radio source sample is then divided into two classes: radio-loud active galactic nuclei (AGN) and galaxies in which the radio emission is dominated by star formation. The division is based on the location of a galaxy in the plane of 4000{AA} break strength versus radio luminosity per unit stellar mass and provides a sample of 2215 radio-loud AGN and 497 star-forming galaxies brighter than 5mJy at 1.4GHz. A full catalogue of positions and radio properties is provided for these sources.