  We show that for any positive integer $m\ge 1$,
  $m$-relator quotients of the modular group $M = PSL(2,\mathbb{Z})$
generically satisfy a very strong Mostow-type \emph{isomorphism rigidity}. We
also prove that such quotients are generically "essentially incompressible". By
this we mean that their "absolute $T$-invariant", measuring the smallest size
of any possible finite presentation of the group, is bounded below by a
function which is almost linear in terms of the length of the given
presentation. We compute the precise asymptotics of the number $I_m(n)$ of
\emph{isomorphism types} of $m$-relator quotients of $M$ where all the defining
relators are cyclically reduced words of length $n$ in $M$. We obtain other
algebraic results and show that such quotients are complete, Hopfian,
co-Hopfian, one-ended, word-hyperbolic groups.
