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We analyze the singularities of rational inner functions (RIFs) on the unit bidisk and study both when these functions belong to Dirichlet‐type spaces and when their partial derivatives belong to Hardy spaces. We characterize derivative    H  p    membership purely in terms of contact order, a measure of the rate at which the zero set of an RIF approaches the distinguished boundary of the bidisk. We also show that derivatives of RIFs with singularities fail to be in    H  p    for    p  ⩾   3  2and that higher non‐tangential regularity of an RIF paradoxically reduces the    H  p    integrability of its derivative. We derive inclusion results for Dirichlet‐type spaces from derivative inclusion for    H  p   . Using Agler decompositions and local Dirichlet integrals, we further prove that a restricted class of RIFs fails to belong to the unweighted Dirichlet space.

Derivatives of rational inner functions: geometry of singularities and integrability at the boundary