Stable rationality of index one Fano hypersurfaces containing a linear space

We prove that a very general complex hypersurface of degree $n+1$ in ${\mathbb{P}}^{n+1}$ containing an $r$-plane with multiplicity $m$ is not stably rational for $n\ge 3$, $m,r>0$ and $n\ge m+r$. We also investigate failure of stable rationality of a very general hypersurface of degree $n+1$ in ${\mathbb{P}}^{n+1}$ admitting several isolated ordinary double points.