  We establish a quasi-monotonicity formula {for an intrinsic frequency
function related to solutions to} thin obstacle problems with zero obstacle
driven by quadratic energies with Sobolev $W^{1,p}$ coefficients, with $p$
bigger than the space dimension. From this we deduce several regularity and
structural properties of the corresponding free boundaries at those
distinguished points with finite order of contact with the obstacle. In
particular, we prove the rectifiability {and the local finiteness of the
Minkowski content} of the whole free boundary in the case of Lipschitz
coefficients.
