  We consider the scheme $X_{r,d,n}$ parametrizing $n$ ordered points in
projective space $\mathbb{P}^r$ that lie on a common hypersurface of degree
$d$. We show that this scheme has a determinantal structure and we prove that
it is irreducible, Cohen-Macaulay, and normal. Moreover, we give an algebraic
and geometric description of the singular locus of $X_{r,d,n}$ in terms of
Castelnuovo-Mumford regularity and $d$-normality. This yields a
characterization of the singular locus of $X_{2,d,n}$ and $X_{3,2,n}$.
