In his "Murphy's Law" paper, Vakil showed that every "singularity type" (with a precise meaning) occurs on certain Hilbert schemes; for instance, the Hilbert scheme of nonsingular curves in projective space. He also gives a method for constructing such singularities; however, the process to construct even, say, a singularity of nodal type would be extremely involved.
[As I understand it, one would have to blow up a plane at something like twenty points (conservative estimate), then take a certain eight-fold cover, then find an appropriate line bundle and take six "sufficiently general" sections,...]
For a smooth variety $X$, let $H_X$ denote its Hilbert scheme. A point of $H_X$ corresponds to a subscheme $V$ of $X$. I am interested in cases in which $V$ is also smooth. [Edit: I am also requiring that $H^1(T_X) = 0$, i.e., that the ambient variety $X$ admit no infinitesimal deformations. To put it another way, the complex structure on the smooth manifold $X$ cannot be deformed. This holds in particular if $X=\mathbb P^n$.] Certainly, explicit examples of such pairs $(V,X)$ corresponding to singular points of $H_X$ have been described; however, in all the (two) very few examples I have seen, the technique is to show that $V$ is contained in an irreducible component of $H_X$ that is generically non-reduced.
Can anyone give explicit examples of a smooth projective variety $X$, X$ [such that $H^1(X,T_X) = 0$], together with a smooth subvariety $V$, such that the point $[V]\in H_X$ is both singular and reduced? [A method of constructing explicit examples will not suffice unless you can show, by example, that this method is actually practical to carry out.]