  We combine $SU(5)$ Grand Unified Theories (GUTs) with $A_4$ modular symmetry
and present a comprehensive analysis of the resulting quark and lepton mass
matrices for all the simplest cases. Classifying the models according to the
representation assignments of the matter fields under $A_4$, we find that there
are seven types of $SU(5)$ models with $A_4$ modular symmetry. We present 53
benchmark models with the fewest free parameters. The parameter space of each
model is scanned to optimize the agreement between predictions and experimental
data, and predictions for the masses and mixing parameters of quarks and
leptons are given at the best fitting points. The best fit predictions for the
leptonic CP violating Dirac phase, the lightest neutrino mass and the
neutrinoless double beta decay parameter when displayed graphically are
observed to cover a wide range of possible values, but are clustered around
particular regions, allowing future neutrino experiments to discriminate
between the different types of models.
