We calculate the radio-frequency spectrum of a trapped cloud of cold bosonic atoms in an optical lattice. By using random phase and local-density approximations we produce both trap-averaged and spatially resolved spectra, identifying simple features in the spectra that reveal information about both superfluidity and correlations. Our approach is exact in the deep Mott limit and in the dilute superfluid when the hopping rates for the two internal spin states are equal. It contains final state interactions, obeys the Ward identities (and the associated conservation laws), and satisfies the f-sum rule. Motivated by earlier work by Sun, Lannert, and Vishveshwara [Phys. Rev. A 79, 043422 (2009)], we also discuss the features that arise in a spin-dependent optical lattice. © 2010 The American Physical Society.