ABSTRACT Only a handful of galaxies in the local Universe appear to be very young. We estimate the fraction of very young galaxies (VYGs), defined as those with more than half their stellar masses formed within the last Gyr. We fit non-parametric star formation histories (SFHs) to ∼280 000 galaxy spectra from a flux- and volume-limited subsample of the Main Galaxy Sample (MGS) of the SDSS, which is also complete in mass-to-light ratio, thus properly accounting for passive galaxies of a given mass. The VYG fractions decrease with increasing galaxy stellar mass, from ∼50 per cent at $m = 10^8\, {\rm M}_{⊙ }$ to ∼0.1 per cent at $m = 10^{11.5}\, {\rm M}_{⊙ }$, with differences of up to 1 dex between the different spectral models used to estimate the SFH and on how we treat aperture effects. But old stellar populations may hide in our VYGs despite our conservative VYG sample built with galaxies that are globally bluer than within the region viewed by the SDSS fibre. The VYG fractions versus mass decrease more gradually compared to the Tweed et al. predictions using analytical and semi-analytical models of galaxy formation, but agree better with the SIMBA hydrodynamical simulation. These discrepancies highlight the usefulness of VYGs in constraining the strong uncertainties in both galaxy formation models and spectral modelling of galaxy SFHs. Given the lognormal cosmic SFH, these mean VYG fractions suggest that galaxies above $10^8\, \rm M_⊙$ undergo at most four major starbursts on average.