Matrix Clark measures appear naturally in the study finite rank perturbations, as well as in the theory of de Branges--Rovnyak spaces. Both approaches give the same measure if the characteristic function $\theta$ satisfies $\theta(0)=0$; in the general case the measures differ by constant normalizing factors.

In the talk I describe recent results about fine properties of the matrix Clark measures, especially of the singular parts of such measures. The notion of the Carath\'{e}odory angular derivative will be introduced for the matrix Clark measures, and connection with point masses will be discussed.

While the results generalize known statements about scalar Clark measures, the generalization is far from trivial: the ideas of directionality play an important role and present a major obstacle in the matrix case.

The talk is based on a joint work with C.~Liaw and R.~T.~W.~Martin.