One emerging trend seems to be that the category of sheaves of sets on the site of smooth manifolds (also known as the category of generalized manifolds) is the right category of manifold-like objects. In particular, it includes all sorts of infinite-dimensional manifolds, such as Banach and Fréchet manifolds as full subcategories. It also contains many other categories of manifold-like objects, such as diffeological spaces, as full subcategories. Also, this category can be generalized nicely to higher smooth homotopy types, e.g., smooth stacks = smooth homotopy 1-types, which constantly pop up even if you're studying ordinary differential geometry.

As an example, one can cite the following result: Consider the smooth stack B^∇G of smooth principal G-bundles with connection and the smooth set Ω of differential forms. The set of maps B^∇G→Ω turns out to be canonically isomorphic to the algebra of Ad-invariant polynomials on the Lie algebra of G. Thus one recovers Chern-Weil theory in a very natural way. See the recent paper by Freed and Hopkins for details: http://arxiv.org/abs/1301.5959. I don't think this result can be obtained in any other model of manifold-like objects, because other models do not allow for spaces like B^∇G and Ω.

One emerging trend seems to be that the category of sheaves of sets on the site of smooth manifolds (also known as the category of generalized manifolds) is the right category of what one might call smooth sets. (Here we no longer restrict our attention to spaces that look the same at every point, and in fact we have spaces that have no points at all.) In particular, it includes all sorts of infinite-dimensional manifolds, such as Banach and Fréchet manifolds as full subcategories. It also contains many other categories of smooth objects, such as diffeological spaces, as full subcategories. Also, this category can be generalized nicely to higher smooth homotopy types, e.g., smooth stacks = smooth homotopy 1-types, which constantly pop up even if you're studying ordinary differential geometry.

As an example, one can cite the following result. Consider the smooth stack B^∇G of smooth principal G-bundles with connection and the smooth set Ω of differential forms. The set of maps B^∇G→Ω turns out to be canonically isomorphic to the algebra of Ad-invariant polynomials on the Lie algebra of G. Thus one recovers Chern-Weil theory in a very natural way. See the recent paper by Freed and Hopkins for details: http://arxiv.org/abs/1301.5959. I don't think this result can be obtained in any other model of smooth objects, because other models do not allow for spaces like B^∇G and Ω.