To add to Daniel's answer, the precise density of hereditary classes has been extensively investigated in combinatorics.
For a class $C$ of structures, the unlabelled slice $C_n$ is the set of isomorphism classes of structures in $C$ that have $n$ vertices.
The (unlabelled) speed of a class $C$ of structures is $|C_n|$.
Denote the class of graphs by $G$.
The question is asking whether $\lim_{n\to \infty} |Q_n|/|G_n| = 1$ for any hereditary class of graphs $Q$.
Since the limit is always 0 for hereditary $Q$, a fundamental question is then how the function $|Q_n|$ itself behaves.
Let $p(n)$ denote the number of integer partitions, where $p(n) = 2^{\Theta(\sqrt{n})}$.
It turns out that the unlabelled speed "jumps": either $|Q_n|$ is polynomially bounded, or otherwise $|Q_n| = \Omega(p(n))$.
- József Balogh, Béla Bollobás, Michael Saks and Vera T. Sós, The unlabelled speed of a hereditary graph property, Journal of Combinatorial Theory, Series B, 99 9–19, 2009.
doi:10.1016/j.jctb.2008.03.004 (preprint)