Given a coherent $\mathcal{D}\_X$-Module $M$, one can assign to it its characteristic variety 
$$ch(M)\subseteq T^\*X$$
or one could look at its support 
$$supp(M)\subseteq X$$
as an $\mathcal {O}\_X$ module. Is there a relation between these two spaces?


Edit: Is it true, that the characteristic variety of a holonomic $\mathcal{D}\_X$-Module is the conormal space of its support?



Relation between characteristic variety and support of D-Module
I'm not an expert either but, at least for holonomic D-modules, the relation you're looking for should be 
$$ 
 supp(M) = ch(M) \cap T^\*\_X X
$$ 
where $T^\*\_XX$ is the zero section of the cotangent bundle identified with $X$.


You can check it in the basic examples: $X = \mathbb{C}$, $M$ the trivial bundle or the $\delta$-module at $x\in X$. The corresponding $ch(M)$ is $T^\*\_XX$ or $T^\*\_xX$. The general case should be an easy exercise in linear algebra. 


