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?
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.
I'm not a $D$-module person. I'm hoping someone else can give a slightly more insightful explanation. (It looks like YBL gives a clear picture of the holonomic case.)
By definition $Ch(M)$ is the support of the associated graded of $M$, for a suitable filtration, so it should lie in the preimage of the support of $M$. However, in many interesting cases the inclusion would be strict. If $M$ is holonomic, $Ch(M)$ is Lagrangian, so it wouldn't coincide with the preimage of $supp(M)$. The simplest case where this happens is when $M$ is a flat connection, then $Ch(M)$ is the zero section of $T^*X$.
Continuation: Perhaps it's worthwhile making this a little more explicit. The simplest example is $X=\mathbb{A}^n$. Then (the global sections of) $D_X$ is the Weyl algebra with generators $x_1,\ldots, x_n, \partial_1,\ldots, \partial_n$ with commutation relations $[x_i,\partial_j]= \delta_{ij}$. If we force these to commute, by passing to the associated graded with respect to the filtration by order of operators, we obtain the polynomial ring in $2n$ variables or in other words the coordinate ring of $T^*\mathbb{A}^n$. Any finitely generated $D_X$-module $M$, carries a (noncanonical) compatible filtration, so can define $Ch(M)= Supp(GrM)\subset T^*\mathbb{A}^n$ (it is independent of the filtration). Now let $M= D/\sum D\partial_i$, which corresponds to $\mathcal{O}_X$. I'll omit the details, but one can see that $Ch(M) = V(\partial_1,\ldots \partial_n)$ (the zero section), and $\pi^*supp(M) = T^*\mathbb{A}^n$ where $\pi:T^*\mathbb{A}^n\to \mathbb{A}^n$.
For a nonholonomic example, take $M= D_X$. Then $Ch(M)= \pi^{-1}supp(M) = T^*\mathbb{A}^n$.
