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 $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$ parametrizing symmetric subschemes of $\mathbb{C}^n$ whose coordinate rings, as $S_n$-modules, are isomorphic to a given representation $ρ$. In the case that $ρ= M^λ$ is a permutation module corresponding to certain special types of partitions $λ$ of $n$, we prove that $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$ is irreducible or even smooth. We also prove irreducibility whenever $\dim ρ\leq 2n$ 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 $\mathrm{Hilb}_ρ^{S_n}(\mathbb{C}^n)$. A central tool is the combinatorial theory of higher Specht polynomials.