A standard reference is:
F. Sergeraert "Un theoreme de fonctions implicites sur certains espaces de Frechet et quelques applications," Ann. Sci. Ecole Norm. Sup. (4) 5 (1972), 599-660.
This isn't a stratification of the space of maps $M \to \mathbb R$ but it is a stratification of an infinite co-dimension subspace of the space of all smooth maps $M \to \mathbb R$. It's a relatively popular stratification to use among geometric topologists, in that it produces Cerf theory. Rubinstein, Hong and McCullough use it in their work on the homotopy-type of $\operatorname{Diff}(L_{p,q})$. (which is how I learned of it)
Is this roughly what you're looking for?
