  For an essentially small triangulated category $\mathcal{T}$, we introduce
the notion of prime thick subcategories and define the spectrum of
$\mathcal{T}$, which shares the basic properties with the spectrum of a tensor
triangulated category introduced by Balmer. We mainly focus on triangulated
categories that appear in algebraic geometry such as the derived and the
singularity categories of a noetherian scheme $X$. We prove that certain
classes of thick subcategories are prime thick subcategories of these
triangulated categories. Furthermore, we use this result to show that certain
subspaces of $X$ are embedded into their spectra as topological spaces.
