If $M,N,S$ are manifolds ($M$ and $S$ compact) and $g:Sto N$ is a smooth map, then if we endow $C^infty(M,N)$ with the strong topology we get that

$$pitchfork(M,N;g):={fin C^infty(M,N),;,fpitchfork g}$$

is dense and open in $C^infty(M,N)$ (see Hirsch, Differential Topology, and Kosinski, Differential Manifolds). This indicates that $pitchfork(M,N;g)$ would fit to be the $0$-stratum of some stratification on $C^infty(M,N)$.

I would be interested to know if someone already established such stratification, with a particular interest in the description of the $1$-stratum he or she came with. I am also very interested if the article discusses how a generic path $f_tin C^infty(M,N)^{(0,1)}$ behaves with respect to this stratification. Thanks for your help!