We study a Signorini type thin obstacle problem near a fixed boundary, that is on a part of the boundary we "clamp down" the solution. We show that in general the solutions are at least C^(0,1/2) regular and that regularity is sharp. However, we are able to prove that near the actual points of contact of the free boundary with the fixed one the solution had to be at least C^(1,1/2) regular, while away from it the global solutions are homogeneous with discrete homogeneity degrees 1/2, 3/2, 5/2, ... Joint work with A. Petrosyan.