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