Interesting (non) examples of singular support

Question

I'm trying to better understand singular support of sheaves on smooth manifolds---to this end: What are examples of conical subsets of $T^*X$ that cannot arise as the singular support of a sheaf on $X$?

As an example, if I have a line $ L $ in $X = R^2$, and I choose a covector field along $L$ which is not conormal to that line, is it possible to construct a sheaf whose singular support in $X$ is given by this covector field along $ L $ (and is zero outside of $ T^*X|_L$ )?

Alternatively, what are some surprising examples of conical subsets one can obtain as the singular support of a sheaf (or complexes thereof)? I would consider conormals to smooth subsets as "obvious" conical subsets one can obtain.

Answer 1

The singular support of a sheaf is always coisotropic (Theorem 6.5.4 in Kashiwara and Schapira).

In the example of the 2-dimensional conic subset of $T^\ast \mathbb R^2$ defined by a covector field along a line, such a subset will not define a coisotropic submanifold unless the covector is conormal to the line. Thus it cannot be the singular support of a sheaf.

The singular support of a (weakly) constructible sheaf is lagrangian, and contained in (but not necessarily equal to) the union of the conormal bundles to the strata.

However, if you look at sheaves that are far from constructible (like the sheaf of smooth functions $C^\infty_X$ for example), the singular support will be the whole of $T^\ast X$. Not sure if this counts as surprising...