The free boundary for the Signorini problem in R n+ 1 is smooth outside of a degenerate set, which can have the same dimension (n− 1) as the free boundary itself. In [15] it was shown that generically, the set where the free boundary is not smooth is at most (n− 2)-dimensional. Our main result establishes that, in fact, the degenerate set has zero H n− 3− α 0 measure for a generic solution. As a by-product, we obtain that, for n+ 1≤ 4, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in R 3 and R 4 for the thin obstacle problem.