Preimage of singular points of smooth map between manifolds Given a smooth ($C^{\infty}$) map $\phi: V \rightarrow SU(n)$ where $V$ is a (finite dim, real) vector space (of potentially very large dimension) and $SU(n)$ is the special unitary Lie group, what can be said about the  of singularities of this map?
It is know that the preimage $\phi^{-1}(v)$ of a regular value $v$ is a submanifold of $V$. What is known about the preimage of a singular value? Is this also a manifold in this case? What about the preimage of all singular points?
Do the singular values form a manifold? I known they are a null set by Sard's theorem but this does not use anything specific to $SU(n)$. Does the target space being $SU(n)$ help us to say anything more about the singular points?
If we further know that the map $\phi$ is onto does this affect things?
 A: Just a partial answer.

It is know that the preimage $ϕ^{-1}(v)$ of a regular value v is a submanifold of V. What is known about the preimage of a singular value? Is this also a manifold in this case?

There is Transversality theorem, which is generalisation of known fact about regular values that you mentioned. Narasimhan in his book "Analysis on real and complex manifolds" states it in following manner:

where trasversal means:

This theorem puts additional condtions on $\phi,$ but I believe still it is a good one.
A: I don't think the target space being $SU(n)$ gives you much information:
Preimages of singular values can be extremely nasty. Any closed subset of $\mathbb{R}$ is the set of singular points of a function $\mathbb{R}\rightarrow \mathbb{R}$. Embedding $\mathbb{R}$ into $SU(n)$ gives you nasty subsets of $\mathbb{R}$. This idea generalizes to higher dimensional vector spaces (although I don't believe any closed subset is the critical set of a smooth function). 
Consider the function $e^{1/x} \sin(1/x)$. The critical values are a null set (by sard), but do have an accumulation point $0$, so they do not form a manifold.
More information about $\phi$ will of course give you more information about the critical set: e.g. if $\phi:\mathbb{R}\rightarrow \mathbb{R}$ is Morse, we know the that the critical set is discrete, and we have a nice local model of the function around its critical points.
