I've been reading Paul Balmer's paper about constructing a "spectrum of prime ideals" on (essentially small) tensor triangulated category in order to then classify thick subcategories. This is all done to generalize work done in various fields throughout mathematics (e.g. Devinatz, Hopkins, and Smith's work in stable homotopy theory, and Pevtsova and Friedlander's work in finite group schemes). So the classic examples of tensor trianulated categories that Balmer talks about are the category of spectra of some space, the category of $G$-modules for some finite group scheme $G$, or the perfect derived category associated to a (topologically Noetherian) scheme (this is related to Thomalson's work reconstructing a scheme from the aforementioned category).