When defining the normality of a scheme in the book The Geometry of Syzygies, Eisenbud says that there are just a few facts that are known for their bounds.

Given $X \subset \mathbb{P}^r$, we say that $X$ is $d$-normal if the map $\mathrm{H}^0(\mathbb{P}^r, \mathcal{O}_{\mathbb{P}^r}(d)) \to \mathrm{H}^0(\mathbb{P}^r, \mathcal{O}_{X}(d))$ is surjective.

I tried to search for examples of $d$-normal schemes, and indeed it is very hard to find them, even when considering sets of simple points in the plane I was not able of giving examples different from those mentioned in the book. Does someone know a reference for this? I am particularly interested in sets of points.