A new method for spectroscopic bulge–disc decomposition is presented, in which the spatial light profile in a two-dimensional spectrum is decomposed wavelength by wavelength into bulge and disc components, allowing separate one-dimensional spectra for each component to be constructed. This method has been applied to observations of a sample of nine lenticular galaxies (S0s) in the Fornax Cluster in order to obtain clean high-quality spectra of their individual bulge and disc components. So far this decomposition has only been fully successful when applied to galaxies with clean light profiles, without contamination from dust lanes, etc. This has consequently limited the number of galaxies that could be separated into bulge and disc components. The Lick index stellar population analysis of the component spectra reveals that in those galaxies where the bulge and disc could be distinguished, the bulges have systematically higher metallicities and younger stellar populations than the discs. This correlation is consistent with a picture in which S0 formation comprises the shutting down of star formation in the disc accompanied by a final burst of star formation in the bulge. Similarly, a trend was found to exist whereby galaxies with younger stellar populations have higher metallicities. The variation in spatial-fit parameters with wavelength also allows us to measure approximate colour gradients in the individual components. Such gradients were detected separately in both bulges and discs, in the sense that redder light is systematically more centrally concentrated in all components. However, a search for radial variations in the absorption line strengths determined for the individual components revealed that, although they can be sensitively detected where present, they are absent from the vast majority of S0 discs and bulges. The absence of gradients in line indices for most galaxies implies that the colour gradient cannot be attributed to age or metallicity variations and is therefore most likely associated with varying degrees of obscuration by dust.