TITLE: What are examples of D-modules that I should have in mind while learning the theory?
QUESTION [22 upvotes]: I've been reading about D-modules this summer in preparation for a learning seminar on intersection cohomology. Unfortunately, many of the ideas are not sticking while I learn about the theory. What are examples of D-modules, which demonstrate

monodromy
D-modules on nontrivial spaces, such as projective curves or surfaces
D-modules with support on a singular space (so I can apply Kashiwara's theorem to see what D-modules can say about singularities)
D-modules on families of varieties over a fixed base

Also, what are some classical systems of differential equations I should be looking at while doing computations? I'm not very familiar with the classical theory.

REPLY [3 votes]: I will edit this question as I learn more, but here's one useful example: Consider the $\mathcal{D}_{\mathbb{A}^1}$-module
$$
\frac{\mathcal{D}_{\mathbb{A}^1}}{\mathcal{D}_{\mathbb{A}^1}(t\partial_t - \beta)}
$$
is a local system on $\mathbb{C}^*$ with monodromy $\text{exp}(2\pi i \beta)$. Apparently this is a useful example for showing the definition of nearby cycles in mixed hodge modules is the right one.