In 2005, Balmer introduced the notion of a prime thick tensor ideal for a tensor triangulated category T as an analogous concept to a prime ideal of a commutative ring. Using prime thick tensor ideals, Balmer established the epoch-making theory so-called the tensor-triangular geometry which allows us to study T by commutative-algebraic/algebro-geometric approaches. On the other hand, recently I have introduced the notion of prime thick subcategories to develop a similar theory to the tensor-triangular geometry for tensor triangulated categories without tensor structures. In this talk, we study prime thick subcategories of the perfect derived category D^perf(X), the bounded derived category D^b(X), and the singularity category D^sg(X) of a noetherian scheme X. Especially, we give a characterization of a point x of X to be a complete intersection or a hypersurface in terms of prime thick subcategories of such derived categories.