Consider $B$ a finite type integral scheme over $\mathbb{Z}$ with $B(\mathbb{Z})$ infinite. (If you like, take $B$ to be the affine line minus a finite set of $\mathbb{Z}$-points). Let $X \to B$ be a smooth proper family of curves of genus $g > 1$. Assume the $X_b, b \in B(\mathbb{Z})$, for which $X_b(\mathbb{Q}) = \emptyset$, are Zariski-dense in $X$. Must then the proportion of members $X_b$, over $b \in B(\mathbb{Z})$, for which $X_b(\mathbb{Q}) \neq \emptyset$, be equal to $0$? (Variant: the same question with $B(\mathbb{Z})$ replaced by $B(\mathbb{Q})$.)

An example. Let $f \in \mathbb{Z}[x]$ be irreducible of degree $> 4$. Do the integers $N$ for which $f(x) = Ny^2$ has a rational solution, have density zero? Variant: let $N$ range over the primes, or over the squarefrees.

This is motivated by a brief discussion with François Brunault here:

http://mathoverflow.net/questions/121056/a-maximalist-scenario-for-the-uniformity-of-mordell-weil-groups .