Simplicial presheaves on cartesian spaces provide a general notion of smooth spaces. We define a corresponding smooth version of the singular complex functor, which maps smooth spaces to simplicial sets. We exhibit this functor as one of several Quillen equivalences between the Kan-Quillen model category of simplicial sets and a motivic-style $\mathbb{R}$-localisation of the (projective or injective) model category of smooth spaces. These Quillen equivalences and their interrelations are powerful tools: for instance, they allow us to give a purely homotopy-theoretic proof of a Whitehead Approximation Theorem for manifolds. Further, we provide a functorial fibrant replacement in the $\mathbb{R}$-local model category of smooth spaces. This allows us to compute the homotopy types of mapping spaces in this model category in terms of smooth singular complexes. We explain the relation of our fibrant replacement functor to the concordance sheaves introduced recently by Berwick-Evans, Boavida de Brito, and Pavlov. Finally, we show how the $\mathbb{R}$-local model category of smooth spaces formalises the homotopy theory on sheaves used by Galatius, Madsen, Tillmann, and Weiss in their seminal paper on the homotopy type of the cobordism category.