Question

Does anybody know any classification of stable singularities of smooth map $f:\mathbb R^3\to \mathbb R^4$?

It is clear that there are singularities which look like intersection of 2 (or 3 or 4) hyperplanes in $\mathbb R^4$. But there are other ones.

Another type of stable singularity can be produced the following way: let $f$ maps $\mathbb R^3$ such way that for $x\in D^2$ $f(x)=f(-x)$ for $x\in \mathbb R^2, x=(x_1,x_2,0,0)$.

Answer 1

The codimension of $\Sigma^2$-points in the source for a codimension $k$ map is $2(k+2)$, in this case that's $6>3$, so you only get combinations of Morin and regular points. From the Morins, the cusp has codimension $2(k+1)=4$, so you have no cusps either and need to keep track only of the regular points (codimension $k=1$ in the target) and fold points (canonical form $(x,y,z) \mapsto (x^2,xy,y,z)$, codimension $2k+1=3$ in the target). So, you have the regular points with multiplicity 1 to 4, the fold curves and fold curves intersecting transversally a regular "branch", and that's all. What you wrote does not look like a stable singularity to me.