Non-parametric sets of regular realizations over number fields

Given a number field k, we show that, for many finite groups G, all the Galois extensions of k with Galois group G cannot be obtained by specializing any given finitely many Galois extensions E/k(T) with Galois group G and E/k regular. Our examples include abelian groups, dihedral groups, symmetric groups, general linear groups over finite fields, etc. We also provide a similar conclusion while specializing any given infinitely many Galois extensions E/k(T) with Galois group G and E/k regular of a certain type, under a conjectural “uniform Faltings' theorem”.