How explicit are the model structures for various categories of spectra?
Naive, symmetric and orthogonal spectra are obtained via left Bousfield localization of model structures with explicit generating (acyclic) cofibrations, so we have explicit generating cofibrations for them.
I'm thinking it's too much to ask for explicit generating acyclic cofibrations, but it would be nice to at least have a pseudo-generating set -- i.e. an explicit set of generating acyclic cofibrations, lifting against which characterizes the fibrant objects and fibrations between fibrant objects.
Questions: Let $\mathcal C$ be a model category modeling spectra, (e.g. naive, symmetric, orthogonal, EKMM, combinatorial...)
Are explicit generating cofibrations available for $\mathcal C$?
How about explicit generating acyclic cofibrations?
If not (2), how about an explicit pseudo-generating set of acyclic cofibrations?
If not (3), is there at least an explicit description of the fibrant objects via lifting properties?