  We provide evidence for this conclusion: given a finite Galois cover $f: X
\rightarrow \mathbb{P}^1_\mathbb{Q}$ of group $G$, almost all (in a density
sense) realizations of $G$ over $\mathbb{Q}$ do not occur as specializations of
$f$. We show that this holds if the number of branch points of $f$ is
sufficiently large, under the abc-conjecture and, possibly, the lower bound
predicted by the Malle conjecture for the number of Galois extensions of
$\mathbb{Q}$ of given group and bounded discriminant. This widely extends a
result of Granville on the lack of $\mathbb{Q}$-rational points on quadratic
twists of hyperelliptic curves over $\mathbb{Q}$ with large genus, under the
abc-conjecture (a diophantine reformulation of the case
$G=\mathbb{Z}/2\mathbb{Z}$ of our result). As a further evidence, we exhibit a
few finite groups $G$ for which the above conclusion holds unconditionally for
almost all covers of $\mathbb{P}^1_\mathbb{Q}$ of group $G$. We also introduce
a local-global principle for specializations of Galois covers $f: X \rightarrow
\mathbb{P}^1_\mathbb{Q}$ and show that it often fails if $f$ has abelian Galois
group and sufficiently many branch points, under the abc-conjecture. On the one
hand, such a local-global conclusion underscores the "smallness" of the
specialization set of a Galois cover of $\mathbb{P}^1_\mathbb{Q}$. On the other
hand, it allows to generate conditionally "many" curves over $\mathbb{Q}$
failing the Hasse principle, thus generalizing a recent result of Clark and
Watson devoted to the hyperelliptic case.
