  A symmetric ideal is an ideal in a polynomial ring which is stable under all
permutations of the variables. In this paper we initiate a global study of
zero-dimensional symmetric ideals. By this we mean a geometric study of the
invariant Hilbert schemes $\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$
parametrizing symmetric subschemes of $\mathbb{C}^n$ whose coordinate rings, as
$S_n$-modules, are isomorphic to a given representation $\rho$. In the case
that $\rho = M^\lambda$ is a permutation module corresponding to certain
special types of partitions $\lambda$ of $n$, we prove that
$\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$ is irreducible or even smooth. We
also prove irreducibility whenever $\dim \rho \leq 2n$ and the invariant
Hilbert scheme is non-empty. In this same range, we classify all homogeneous
symmetric ideals and decide which of these define singular points of
$\mathrm{Hilb}_{\rho}^{S_n}(\mathbb{C}^n)$. A central tool is the combinatorial
theory of higher Specht polynomials.
