  We investigate the consequences of natural conjectures of Montgomery type on
the non-vanishing of Dirichlet $L$-functions at the central point. We first
justify these conjectures using probabilistic arguments. We then show using a
result of Bombieri, Friedlander and Iwaniec and a result of the author that
they imply that almost all Dirichlet $L$-functions do not vanish at the central
point. We also deduce a quantitative upper bound for the proportion of
Dirichlet $L$-functions for which $L(\frac 12,\chi)=0$.
