  We use a $K$-theory recipe of Thomason to obtain classifications of
triangulated subcategories via refining some standard thick subcategory
theorems. We apply this recipe to the full subcategories of finite objects in
the derived categories of rings and the stable homotopy category of spectra.
This gives, in the derived categories, a complete classification of the
triangulated subcategories of perfect complexes over some commutative rings. In
the stable homotoy category of spectra we obtain only a partial classification
of the triangulated subcategories of the finite $p$-local spectra. We use this
partial classification to study the lattice of triangulated subcategories. This
study gives some new evidence to a conjecture of Adams that the thick
subcategory $\C_2$ can be generated by iterated cofiberings of the Smith-Toda
complex. We also discuss several consequences of these classification theorems.
