Can the constant rank theorem for smooth manifolds be generalized to nonconstant rank? The constant rank theorem says that
if $f\colon M→N$ is a smooth map whose rank equals some fixed $k≥0$ at any point of $M$, then, locally with respect to $M$ and $N$, the map $f$ assumes the easiest possible form:
$f(x_1,…,x_k,…,x_m)=(x_1,…,x_k,0,…,0)$.
I am interested in a more general situation: the rank of $f$ is at most $k$
at any point of $M$ and can actually drop below $k$ at some points.
The classification of local forms for such maps seems to be too difficult.
However, in my case one can replace the map f with any map homotopic to it
via a homotopy $M×[0,1]→N$ that is itself a smooth map of rank at most $k$.
Thus, one can inquire about the possible local forms of smooth maps
of rank at most $k$ up to smooth homotopy of rank at most $k$.
I am mostly interested in the case $k < \operatorname{dim} M$.
Such a problem could presumably be solved using the techniques of catastrophe theory,
but my knowledge of this area is virtually nonexistent.
Is it possible to write down a finite list of local forms of smooth maps
of rank at most $k$ considered up to a smooth homotopy of rank at most $k$?
This is motivated by the following question,
which arises in connection to holonomy and parallel transport:
given some $k<m$, can any smooth map $S^m→\mathbb{R}^n$ of rank at most $k$
be extended to a smooth map $D^{m+1}→\mathbb{R}^n$ of rank at most $k$?
 A: Your motivating question has a negative answer:  Consider the inclusion $\iota:S^m\to\mathbb{R}^{m+1}$, which has rank at most $m$.  If there were a smooth extension $f:D^{m+1}\to\mathbb{R}^{m+1}$ of rank at most $m$ everywhere, then all the points of $D^{m+1}$ would be critical points for $f$ and hence, by Sard's Theorem, the set of critical values, i.e., the entire image $f(D^{m+1})$, would be a set of measure zero in $\mathbb{R}^{m+1}$.  In particular, it would have to miss some point $p$ in the interior of $D^{m+1}$.  However, since $p$ is in the interior, the inclusion $\iota:S^m\to\mathbb{R}^{m+1}\setminus\{p\}$ is not null homotopic, and hence there cannot be any extension $f:D^{m+1}\to \mathbb{R}^{m+1}\setminus\{p\}$.
If you want a map of rank less than $m$ (which is the dimension of $S^{m}$), just take the Hopf map $\pi:S^3\to S^2\subset\mathbb{R}^3$, which has rank $2<3$ everywhere.  Then, by the same Sard's Theorem argument as above, coupled with the fact that $\pi$ generates $\pi_3(S^2)\simeq\mathbb{Z}$, it follows that $\pi$ cannot be extended to a smooth map $f:D^4\to\mathbb{R}^3$ of rank at most $2$ everywhere.  
As for your boldface question, I am not really clear on what you are asking.  For example, wouldn't a positive answer to your question mean that there is a finite list of 'local forms up to homotopy' for smooth maps between two manifolds of the same dimension?  I think that this is known to be false, even up to homotopy, when the dimension of the manifolds is sufficiently large.  There is also the complication caused by the existence of homeomorphic but not diffeomorphic manifolds.  
