We study the topological and geometric aspects of the moduli space that parametrizes smooth fibration structures on a given manifold. Complementary to the classification problem, we investigate each equivalence class as an infinite-dimensional manifold inheriting structures from the diffeomorphism group. This interacts with the Lie groups of gauge symmetries and of spacetime symmetries on the one hand, and with the the geometric analysis of extrinsic geometric flows on the other hand. In particular, we focus on the quest of minimal deformation retracts (the homotopy “cores”) for such moduli spaces, with which we extract precise information on low-dimensional manifolds. For example, the homotopy and topological types for moduli spaces of circle fibrations on various surfaces and three-manifolds are determined explicitly.