Take a topological manifold $M$ endowed with a smooth differential structure $\mathcal{C}^{\infty}$.

  Suppose one considers a strictly coarser topology that is locally Euclidean for the set $M$ such that the topology is not Hausdorff. Such space can be called a non-Hausdorff topological manifold.

Such non-Hausdorff topological manifold can be assigned a differential structure in a non-standard manner

Can there exist any smooth diffeomorphism $f \in \operatorname{Diff}(M)$ with the differential structure $C^{\infty} $, such that it possesses singularities under the change of the topology to the coarser one?

Oppositely one might start with a non-Hausdorff differentiable manifold, and ask if there might appear a singular point, by choosing a finer topology that results in a standard differentiable manifold?

NOTE: PLEASE avoid the coarsest/finest topology over the manifold.

Take a topological manifold $M$. Suppose one considers a strictly coarser topology than the manifold topology. Can such topology result in a non-Hausdorff topological manifold?

NOTE: PLEASE avoid the coarsest/finest topology over the manifold.