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 suprising examples of conical subsets one can obtain as the singular spport of a sheaf (or complexes thereof)? I would consider conormals to smooth subsets as "obvious" conical subsets one can obtain.

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.