  We say that a subset of $\mathbb{P}^n(\mathbb{R})$ is maximally singular if
its contains points with $\mathbb{Q}$-linearly independent homogenous
coordinates whose uniform exponent of simultaneous rational approximation is
equal to $1$, the maximal possible value. In this paper, we give a criterion
which provides many such sets including Grassmannians. We also recover a result
of the author and Roy about a class of quadratic hypersurfaces.
