The topological Hochschild homology T H H (R) of a commutative S–algebra (E∞ ring spectrum) R naturally has the structure of a commutative R–algebra in the strict sense, and of a Hopf algebra over R in the homotopy category. We show, under a flatness assumption, that this makes the Bökstedt spectral sequence converging to the mod p homology of T H H (R) into a Hopf algebra spectral sequence. We then apply this additional structure to the study of some interesting examples, including the commutative S–algebras k u, k o, t m f, j u and j, and to calculate the homotopy groups of T H H (k u) and T H H (k o) after smashing with suitable finite complexes. This is part of a program to make systematic computations of the algebraic K–theory of S–algebras, by means of the cyclotomic trace map to topological cyclic homology.