The tautological ring of a smooth manifold $M$ is the ring of characteristic classes of smooth oriented fibre bundles with fibre $M$ generated by the generalised Miller-Morita-Mumford classes, which are defined as fibre integrals of characteristic classes of the vertical tangent bundle. I will explain how one can study the tautological ring using rational homotopy theory by constructing characteristic classes of fibrations with fibre $M$ that extend the definition of some MMM-classes. Finally, I will discuss how to extract obstructions to smoothing fibrations with fibre $M$ and in particular for some 4-manifolds.