The free boundary for the Signorini problem in Rn+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 Hn-3-alpha 0measure for a generic solution. As a by-pro duct, we obtain that, for n + 1 <= 4, the whole free boundary is generically smooth. This solves the analogue of a conjecture of Schaeffer in R3 and R4 for the thin obstacle problem.(c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).