*Preprint*

**Inserted:** 9 apr 2021

**Last Updated:** 9 apr 2021

**Year:** 2021

**Abstract:**

In this paper we analyze the singular set in the Stefan problem and prove the following results: The singular set has parabolic Hausdorff dimension at most $n-1$.

The solution admits a $C^\infty$-expansion at all singular points, up to a set of parabolic Hausdorff dimension at most $n-2$.

In $\R^3$, the free boundary is smooth for almost every time $t$, and the set of singular times $\mathcal S\subset \R$ has Hausdorff dimension at most $1/2$.

These results provide us with a refined understanding of the Stefan problem's singularities and answer some long-standing open questions in the field

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