  Let $S$ be a smooth projective surface over $\mathbb{C}$. We study the local
and global geometry of the nested Hilbert scheme of points $S^{[n,n+1,n+2]}$.
In particular, we show that $S^{[n,n+1,n+2]}$ is an irreducible local complete
intersection with klt singularities. In addition, we compute the Picard group
of $S^{[n,n+1,n+2]}$ when $h^1(S,\mathcal{O}_S) = 0$. From the irreducibility
of $S^{[n,n+1,n+2]}$, we deduce irreducibility for four other infinite families
of nested Hilbert schemes. We give the first explicit example of a reducible
nested Hilbert scheme, which allows us to show that $S^{[n_1,\dots,n_k]}$ is
reducible for $k > 22$.
