[Wellpowered](http://ncatlab.org/nlab/show/well-powered+category) means that for every scheme $X$, the subobject lattice of monormophisms $Y \to X$ is essentially small; regularly wellpowered means that for every scheme $X$, the regular subobject lattice of [regular monomorphisms](http://ncatlab.org/nlab/show/regular+monomorphism) $Y \to X$ (being a regular mono means that $Y \to X$ is the equalizer of some pair of maps) is essentially small. Wellpoweredness implies regular wellpoweredness, but not conversely.

Let me sheepishly admit that I ask this question knowing next to nothing about algebraic geometry. My motivation comes from [this MO discussion](https://mathoverflow.net/a/159995/2362) where it was clarified that the category of schemes is [concretizable](http://ncatlab.org/nlab/show/concrete+category) (i.e. admits a faithful functor to $\mathbf{Set}$) if and only if it is regularly wellpowered.

Here's what I know: [This MO question](https://mathoverflow.net/questions/56591/what-are-the-monomorphisms-in-the-category-of-schemes) quotes SGA giving a characterization of monomorphisms *locally of finite type*, but not arbitrary monomorphisms. [This MO question](https://mathoverflow.net/questions/66279/regular-monomorphisms-of-schemes?) indicates that every regular mono is a locally closed immersion. So I would be very happy if someone could tell me whether a scheme can have a large number of locally closed immersions into it.