at.algebraic topology - Serre spectral sequence with spectra

Since the ordinary Serre spectral sequence is about a fibration of spaces, I don't think you can just talk about a "fibration of spectra" and expect that to be a generalization, since the suspension spectrum functor doesn't preserve fibration sequences. However, there is a version of the Serre spectral sequence involving parametrized spectra, which one can think of as a fibration whose base is an ordinary space and whose fibers are spectra. It can be found in section 20.4 of May-Sigurdsson, Parametrized Homotopy Theory.