Is there a known characterization of epimorphisms in the category of schemes?

It is easy to see that a morphism $f : X \to Y$ such that the underlying map $\lvert f\rvert$ is surjective and the homomorphism $f^\# : \mathcal{O}_Y \to f_* \mathcal{O}_X$ is injective, is an epimorphism. But there are other examples, too: if $Y$ is reduced and locally of finite type over a field, the obvious morphism from $X=\bigsqcup_{y\in Y_\text{cl}} \operatorname{Spec}(k(y))$ to $Y$ is an epimorphism (see #8(b) in Mark Haiman's Homework Set 9 for Math 256AB).

If this is is not possible, what about regular, extremal, or effective epimorphisms? Here, again, I know only some examples.

My background is that I want to know if there is a categorical characterization of the spectra of fields in the category of schemes. In the full subcategory of affine schemes, they are characterized by the property: $X$ is non-initial and every morphism from a non-initial object to $X$ is an epimorphism. But I doubt that this characterization takes over to the category of schemes. EDIT: Kevin Ventullo has shown below that the characterization takes over. Thus my original question has been answered (and I wonder if it's appropriate to accept it as an answer). But of course every other hint about the characterization of epimorphisms of schemes is appreciated.