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.