We perform a spectroscopic study to constrain the stellar initial mass function (IMF) by using a large sample of 24 781 early-type galaxies from the Sloan Digital Sky Survey-based Spheroids Panchromatic Investigation in Different Environmental Regions survey. Clear evidence is found of a trend between IMF and central velocity dispersion (σ0), evolving from a standard Kroupa/Chabrier IMF at σ0 ˜ 100 km s-1 towards a more bottom-heavy IMF with increasing σ0, becoming steeper than the Salpeter function at σ0 ≳ 220 km s-1. We analyse a variety of spectral indices, combining gravity-sensitive features, with age- and metallicity-sensitive indices, and we also consider the effect of non-solar abundance variations. The indices, corrected to solar scale by means of semi-empirical correlations, are fitted simultaneously with the (nearly solar-scaled) extended MILES (MIUSCAT) stellar population models. Similar conclusions are reached when analysing the spectra with a hybrid approach, combining constraints from direct spectral fitting in the optical with those from IMF-sensitive indices. Our analysis suggests that σ0, rather than [α/Fe], drives the variation of the IMF. Although our analysis cannot discriminate between a single power-law (unimodal) IMF and a low-mass (≲0.5 M⊙) tapered (bimodal) IMF, robust constraints can be inferred for the fraction in low-mass stars at birth. This fraction (by mass) is found to increase from ˜20 per cent at σ0 ˜ 100 km s-1, up to ˜80 per cent at σ0 ˜ 300 km s-1. However, additional constraints can be provided with stellar mass-to-light (M/L) ratios: unimodal models predict M/L significantly larger than dynamical M/L, across the whole σ0 range, whereas a bimodal IMF is compatible. Our results are robust against individual abundance variations. No significant variation is found in Na and Ca in addition to the expected change from the correlation between [α/Fe] and σ0.