There are a couple different models for spectra, or constructions of the categories of spectra that have the desired properties (homotopically and otherwise). The construction of the Categories of $S$-algebras in EKMM (Rings, Modules, and Algebras in stable homotopy theory by Elmendorf-Mandell-May-Kriz) is one such model. In EKMM, the authors develop a lot of "technology" for their category of spectra. In particular, they construct several spectral sequences and talk about Algebraic K-theory of S-algebras. I am curious to know if such developments have been pursued in the various other models, such as Symmetric Spectra.
All of these models yield the same homotopy category (I believe the paper of Mandell-May-Schwede-Shipley does this, or does a lot of it), but the impression I have gotten is that $S$-algebras have a lot of the technology already developed that one might want in stable homotopy theory. I believe Schwede has worked on computing homotopy groups of Symmetric Spectra, is this a sign that more technology is on its way?
I suppose the above is all background in some sense, so let me clearly state my question(s): Are there developed notions of spectral sequences for other models of spectra? Are there simple reasons why one would or would not expect such developments to occur (other than we already have them for $S$-algebras)?
Also, I don't mean for this to spark a debate about different models. I am just curious.
EDIT: Clearly, I was unclear. I am wondering if people have used, written down, proved results regarding convergence etc, spectral sequences based on other models of spectra. Is it as Tyler points out that all constructions of spectral sequences essentially occur in the homotopy category? That would certainly answer my question. I thought that since computing homotopy groups of Symmetric Spectra (non-naively) was subtle that such constructions might be subtle as well.