  We combine $SO(10)$ 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. We focus on the case where the three
fermion families in the 16 dimensional spinor representation form a triplet of
$\Gamma_3\simeq A_4$, with a Higgs sector comprising a single Higgs multiplet
$H$ in the ${\mathbf{10}}$ fundamental representation and one Higgs field
$\overline{\Delta}$ in the ${\mathbf{\overline{126}}}$ for the minimal models,
plus and one Higgs field $\Sigma$ in the ${\mathbf{120}}$ for the non-minimal
models, all with specified modular weights. The neutrino masses are generated
by the type-I and/or type II seesaw mechanisms and results are presented for
each model following an intensive numerical analysis where we have optimized
the free parameters of the models in order to match the experimental data. For
the phenomenologically successful models, we present the best fit results in
numerical tabular form as well as showing the most interesting graphical
correlations between parameters, including leptonic CP phases and neutrinoless
double beta decay, which have yet to be measured, leading to definite
predictions for each of the models.
