  For every odd integer $n \geq 3$, we prove that there exist infinitely many
number fields of degree $n$ and associated Galois group $S_n$ whose class
number is odd. To do so, we study the class groups of families of number fields
of degree $n$ whose rings of integers arise as the coordinate rings of the
subschemes of $\mathbb{P}^1$ cut out by integral binary $n$-ic forms. By
obtaining upper bounds on the mean number of $2$-torsion elements in the class
groups of fields in these families, we prove that a positive proportion
(tending to $1$ as $n$ tends to $\infty$) of such fields have trivial
$2$-torsion subgroup in their class groups and narrow class groups. Conditional
on a tail estimate, we also prove the corresponding lower bounds and obtain the
exact values of these averages, which are consistent with the heuristics of
Cohen-Lenstra-Martinet-Malle and Dummit-Voight.
  Additionally, for any order $\mathcal{O}_f$ of degree $n$ arising from an
integral binary $n$-ic form $f$, we compare the sizes of
$\mathrm{Cl}_2(\mathcal{O}_f)$, the $2$-torsion subgroup of ideal classes in
$\mathcal{O}_f$, and $\mathcal{I}_2(\mathcal{O}_f)$, the $2$-torsion subgroup
of ideals in $\mathcal{O}_f$. For the family of orders arising from integral
binary $n$-ic forms and contained in fields with fixed signature $(r_1,r_2)$,
we prove that the mean value of the difference $|\mathrm{Cl}_2(\mathcal{O}_f)|
- {2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ is equal to $1$, generalizing
a result of Bhargava and the third-named author for cubic fields. Conditional
on certain tail estimates, we also prove that the mean value of
$|\mathrm{Cl}_2(\mathcal{O}_f)| -
{2^{1-r_1-r_2}}|\mathcal{I}_2(\mathcal{O}_f)|$ remains $1$ for certain families
obtained by imposing local splitting and maximality conditions.
