I am interested in "approximating" smooth maps from a compact smooth manifold $M$ of dim $m$ into $\mathbb R^n,$ for $n>m,$ by "nice" maps, with properties similar to those of Morse functions. Of course, the question is interesting only for $n\leq 2m,$ since every map can be approximated by an immersion for higher $n$. (The case $dim M<n$ is also interesting but less relevant to my needs.)

What work has been done in this area (beyond Sard's Thm)?

Consider for example topologically stable maps. Are the sets of their critical values always finite? What is known about the structure of these maps near their critical values?

Perhaps there is some other dense subset of $C^\infty(M,\mathbb R^n)$ whose elements are functions with finite numbers of critical values and these critical values well understood?