Applications of diffeological spaces to ordinary differential geometry Recently I've been learning more about differential geometry, and I came upon the notion of a diffeological space, which encompasses a number of already known extensions of smooth manifolds or related notions, like Banach and Frechét manifolds, complex and analytic manifolds, but also includes a number of other constructions (like quotients and mapping spaces), making the category of diffeological spaces quite well-behaved and nice to work with.
However, I couldn't find much information about applications of diffeology to "ordinary" differential geometry, and would love to hear about some results in this vein. Have diffeological spaces been used to obtain meaningful results about ordinary manifolds (smooth, complex, analytic, p-adic, etc.), specially for cases in which there's no known proof that does not use diffeology?
One example would be something like using the de Rham cohomology of (or other constructions involving) the diffeological space of diffeomorphisms/symplectomorphisms/smooth maps to prove results about ordinary manifolds.
 A: As far as I am aware, you can find some applications of diffeology "merely" in differential geometry of manifolds in the following list (I am not sure this list is exhaustive):

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*The (internal) tangent space of the diffeomorphism group of a compact manifold at the identity is the space of its vector fields, See [Hector G. Géométrie et topologie des espaces difféologiques. Analysis and geometry in foliated manifolds (Santiago de Compostela, 1994). 1995 Nov 17:55-80.]
or
[J.D. Christensen, E. Wu, Tangent spaces and tangent bundles for diffeological spaces, Cah. Topol. G'{e}om. Diff'{e}r. Cat'{e}g. 57(1) (2016), 3-50.].


*The diffeomorphism group of a Lie foliation, See [Hector G, Macías-Virgós E, Sotelo-Armesto A. The diffeomorphism group of a Lie foliation. InAnnales de l'Institut Fourier 2011 (Vol. 61, No. 1, pp. 365-378).]


*De Rham cohomology of diffeological spaces and foliations, See [Hector G, Macías-Virgós E, Sanmartín-Carbón E. De Rham cohomology of diffeological spaces and foliations. Indagationes Mathematicae. 2011 Aug 1;21(3-4):212-20.]


*The basic de Rham complex of a singular foliation, See [Miyamoto D. The Basic de Rham Complex of a Singular Foliation. International Mathematics Research Notices.]


*Basic forms and orbit spaces: a diffeological approach, See [Karshon Y, Watts J. Basic forms and orbit spaces: a diffeological approach. SIGMA. Symmetry, Integrability and Geometry: Methods and Applications. 2016 Mar 8;12:026.]


*The orientation-preserving diffeomorphism group of $\mathbb{S}^2$ deforms to SO (3) smoothly, See [Li J, Watts JA. The orientation-preserving diffeomorphism group of $\mathbb{S}^2$ deforms to SO (3) smoothly. Transformation Groups. 2011 Jun;16(2):537-53.]


*Smooth Lie group actions are parametrized diffeological subgroups, See [Iglesias-Zemmour P, Karshon Y. Smooth Lie group actions are parametrized diffeological subgroups. Proceedings of the American Mathematical Society. 2012 Feb;140(2):731-9.]


*Differential forms on manifolds with boundary and corners See [Gürer S, Iglesias-Zemmour P. Differential forms on manifolds with boundary and corners. Indagationes Mathematicae. 2019 Sep 1;30(5):920-9.]


*The Geodesics of the 2-Torus See here.


*Every symplectic manifold is a (linear) coadjoint orbit See [Donato P, Iglesias-Zemmour P. Every symplectic manifold is a (linear) coadjoint orbit. Canadian Mathematical Bulletin. 2022 Jun;65(2):345-60.]
