  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% at $m = 10^8\,\rm M_\odot$ to ~0.1%
at $m = 10^{11.5}\,\rm M_\odot$, 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 with $m > 10^8\,\rm M_\odot$ undergo at
most 4 major starbursts on average.
