The generic structure of the singular set in the Stefan problem

We show that, in every space dimension $n\ge2$, the singular set $Σ$ of a generic (in the Baire-category sense) solution of the Stefan problem is countably parabolically $(n-2)$-rectifiable. In particular, $\dim_{\mathrm{par}}(Σ)\le n-2$. This is achieved by proving the following sharp result: in every space dimension, points of the maximal stratum of a generic solution are discrete in space-time.