Stable homotopy of CW R-modules

Modern categories of spectra with strict smash products allow good notions of commutative ring objects and their modules. The homotopy theory of these can be studied using analogues of classical ideas such as cellular techniques, Spanier-Whitehead duality and Adams spectral sequences. I will discuss in detail some connective $2$-local examples ($kO$ and $tmf$) where there is a nice interplay between the new setting an classical stable homotopy. If there is time I will also discuss versions of singularity categories for these.