Are most curves over Q pointless? Fresh out of the arXiv press is the remarkable result of Manjul Bhargava saying that most hyperelliptic curves over $\mathbf{Q}$ have no rational points.  Don Zagier suggests the paraphrase : Most hyperelliptic curves are pointless.
Crucial to the precise mathematical formulation of the statement is a kind of canonical equation for hyperelliptic curves (of a fixed genus) permitting one to define the density of those which have no rational points.
What is the corresponding statement for all curves over $\mathbf{Q}$ ? 
Addendum (2013/09/28) A very nice introduction to the work of Bhargava can be found in How many rational points does a random curve have? by Wei Ho.
 A: I suppose I would say this.  Let h: M_g(Q) -> R be the height function corresponding to the Hodge class.  Let S(N) be the set of points of M_g(Q) of height at most N.  Finally, let P be the image of the projection from M_{g,1}(Q) to M_g(Q).  (i.e. "the set of pointy curves.")  Then one version of the assertion you're looking for would be:
$\lim_{N \rightarrow \infty} \frac{|S(N) \cap P|}{|S(N)|} = 0$.
But as Felipe says, it's possible that asymptotically 100% of the points in $S(N)$ are hyperelliptic, which makes this not very general at all.  So you could go more hardcore and try this.
For all subvarieties X of M_g with infinitely many rational points, either:


*

*$X(\mathbf{Q}) \subset P$ (i.e. every curve parametrized by X has a point)


or


*

*$\lim_{N \rightarrow \infty} \frac{|S(N) \cap P \cap X|}{|S(N) \cap X|} = 0$


That is, either every curve in the family X has a point, or almost none of them do.
Of course, I have no idea whether this is true -- just trying to write something down in the vein of your question which is not "secretly just about hyperelliptic curves."
A: My paper with Bjorn Poonen (which is referenced and discussed in Bjorn's answer to this MO question: Are most cubic plane curves over the rationals elliptic?) has a precise statement for plane curves. You can follow Mike's suggestion in his comment to make a statement for all curves, but this has a problem. Namely, the moduli space of curves of genus $g$ is of general type for $g>22$ (or something like that) so, if you believe Lang's conjecture (or some weakening of it) then there no (or very few) "general" curves of genus $g$ defined over $\mathbb{Q}$, so one expects that most curves of genus $g$ defined over $\mathbb{Q}$ are restricted to rational subvarieties of the moduli space and the biggest one is the hyperelliptic locus, so maybe in some weird sense "most" curves over $\mathbb{Q}$ are hyperelliptic. 
