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We study Clark measures associated with general two-variable rational innerfunctions (RIFs) on the bidisk, including those with singularities, and withgeneral $d$-variable rational inner functions with no singularities. We giveprecise descriptions of support sets and weights for such Clark measures interms of level sets and partial derivatives of the associated RIF. In twovariables, we characterize when the associated Clark embeddings are unitary,and for generic parameter values, we relate vanishing of two-variable weightswith the contact order of the associated RIF at a singularity.

Clark measures for rational inner functions II: general bidegrees and higher dimensions