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In this work, we consider the singular set in the thin obstacle problem withweight $|x_{n+1}|^a$ for $a\in (-1, 1)$, which arises as the local extension ofthe obstacle problem for the fractional Laplacian (a non-local problem). Wedevelop a refined expansion of the solution around its singular points bybuilding on the ideas introduced by Figalli and Serra to study the fineproperties of the singular set in the classical obstacle problem. As a result,under a superharmonicity condition on the obstacle, we prove that each stratumof the singular set is locally contained in a single $C^2$ manifold, up to alower dimensional subset, and the top stratum is locally contained in a$C^{1,\alpha}$ manifold for some $\alpha > 0$ if $a < 0$.  In studying the top stratum, we discover a dichotomy, until now unseen, inthis problem (or, equivalently, the fractional obstacle problem). We find thatsecond blow-ups at singular points in the top stratum are global, homogeneoussolutions to a codimension two lower dimensional obstacle problem (orfractional thin obstacle problem) when $a < 0$, whereas second blow-ups atsingular points in the top stratum are global, homogeneous, and $a$-harmonicpolynomials when $a \geq 0$. To do so, we establish regularity results for thiscodimension two problem, what we call the very thin obstacle problem.  Our methods extend to the majority of the singular set even when no signassumption on the Laplacian of the obstacle is made. In this general case, weare able to prove that the singular set can be covered by countably many $C^2$manifolds, up to a lower dimensional subset.

On the singular set in the thin obstacle problem: higher order blow-ups and the very thin obstacle problem