Let $\mathbb{Z}[x]_n$ be the set of polynomials in $\mathbb{Z}[x]$ of degree at most $n$.
Then, consider some sensible increasing filtration $$A_0 \subset A_1 \subset A_2 \subset \cdots$$
of $\mathbb{Z}[x]_n$ by finite sets, i.e., a sequence of nested, finite sets $A_k$ such that $\bigcup_k A_k = \mathbb{Z}[x]_n$. (One possible sensible option is to take $A_k$ to be the set of polynomials whose coefficients are all in $\{-k, \ldots, k\}$.)
For each $A_k$ and each subgroup $H \leq S_n$ (up to isomorphism), we can ask for the proportion $p_k(H)$ of polynomials in $A_k$ whose Galois group has isomorphism type $H$.
What can be said about the limiting behavior of $p_k(H)$ as $k \to \infty$ that doesn't depend too severely on the filtration $(A_k)$? In particular, is the Galois group of a randomly selected polynomial of degree $< n$ almost surely $S_n$, or more precisely, is $$\lim_{k \to \infty} p_k(S_n) = 1?$$
If not, what is this probability? If it is $0$ or $1$, what can be said about the limiting behavior of $p_k(S_n)$?
What about these questions for other subgroups of $S_n$?
Following links given in the mathoverflow.se link provided by Derek Holt gives answers to some of these:
S. D. Cohen proved that the proportion of polynomials in $\mathbb{Z}[x]_n$ and coefficients all at most $k$ is absolute value for which the Galois group is not $S_n$ is $$\ll \frac{\log k}{\sqrt k},$$ so $$\lim_{k \to \infty} p_k(S_n) = 1 ,$$ and we have a lower bound on convergence.
Remark None of the sources seem, however, to give lower bounds for $p_k(H)$ for proper subgroups $H < S_n$, i.e., a description of the asymptotics of how often particular proper subgroups occur as Galois groups of polynomials in $\mathbb{Z}[x]_n$.