  We present a comprehensive analysis of neutrino mass and lepton mixing in
theories with $A_5$ modular symmetry. We construct the weight 2, weight 4 and
weight 6 modular forms of level 5 in terms of Dedekind eta-functions and Klein
forms, and their decomposition into irreducible representation of $A_5$. We
construct all the simplest models based on $A_5$ modular symmetry, including
scenarios of models with and without flavons in the charged lepton sectors. For
each case, the neutrino masses can be generated through either the Weinberg
operator or the type I seesaw mechanism. We perform an exhaustive numerical
analysis, organising our results in an extensive set of figures and tables.
