  In this note, we give sufficient conditions for the (semi)stability of a
hypersurface $H$ of $\mathbb{P}^N_k$ in terms of its degree $d$, the maximal
multiplicity $\delta$ of its singularities, and the dimension $s$ of its
singular locus. For instance, we show that $H$ is semistable when $d \geq
\delta \min (N+1, s+3)$. The proof relies in particular on Benoist's lower
bound for the dimension of the intersection of the singular locus
$H_{\mathrm{sing}}$ of $H$ with some linear subspace of $\mathbb{P}^N_k$
associated to a one-parameter subgroup $\lambda$ of $\mathrm{SL}_{N+1, k}$, in
terms of the numerical data in the Hilbert-Mumford criterion applied to
$\lambda$ and to an equation $F_H$ of $H$.
