Let $X$ be a connected projective scheme over $\mathbb{C}$ and $E$ a coherent sheaf on $X$. Consider the Quot scheme $\operatorname{Quot}_X(E,P)$ of quotients of $E$ of fixed Hilbert polynomial $P$. As far as I know, this scheme is connected at least in the following cases:

- for Hilbert schemes on $\mathbb{P}^n$ (by Hartshorne)
- for punctual Quot schemes on smooth surfaces (by Fogarty, Ellingsrud–Lehn, Baranovski, etc.).

Are there any other known cases for which $\operatorname{Quot}_X(E,P)$ is connected? Could you give some examples where the connectedness fails? I am particularly interested in higher dimensions or singular surfaces/varieties.