This question is related to <a href="http://mathoverflow.net/questions/157004/for-what-varieties-do-we-have-results-on-the-category-of-singularities">this MO question</a>.

Let $X$ be a projective curve over a field $\mathbb{C}$. We have the bounded derived category of coherent sheaves $D^b_{coh}(X)$ and the derived category of perfect complex $Perf(X)$. It is clear that  $Perf(X)$ is a strictly full triangulated subcategory of  $D^b_{coh}(X)$. Then following <a href="http://arxiv.org/pdf/math/0302304v2.pdf">Orlov 2003</a> we define the triangulated category of singularities of $X$ as the quotient of $D^b_{coh}(X)$ and  $Perf(X)$, i.e.
$$
D_{sg}(X)=D^b_{coh}(X)/Perf(X).
$$

Now is there any study of $D_{sg}(X)$ for the curve $X$? For example does  $D_{sg}(X)$ admit a semiorthogonal decomposition?