Stably equivalent bundles and stable homotopy classes of maps We know that there is a one-to-one correspondence between isomorphism classes of principal $G$-bundles over a base space $M$ and homotopy classes of maps $M \to BG$, where $BG$ is the classifying space of $G$.  
Let us restrict to the case of (real or complex) vector bundles.   What happens to the above correspondence if I consider vector bundles up to stable equivalence and stable homotopy classes of maps $[M,BG]_*$?  Are there any relationships at all?  I'm hoping that there might again be a one-to-one correspondence, but at the moment I have no insight as to whether the correspondence is even well-formed.  
This question arose from some thoughts on how well characteristic classes actually classify vector bundles.  
 A: Real resp. complex $n$-dimensional vector bundles are classified by homotopy classes of maps $M \to BO(n)$ resp. $M \to BU(n)$. Stable real resp. complex vector bundles are classified by homotopy classes (not stable homotopy classes) of maps $M \to BO$ resp. $M \to BU$, where $O$ is the stable orthogonal group and $U$ is the stable unitary group. 
This usage of "stable" is not quite the same as the usage in stable homotopy, although they can be related through the unstable and stable J-homomorphisms. In particular, you will sometimes see the homotopy groups $\pi_n(O)$ resp. $\pi_n(U)$ called the "stable homotopy" of the orthogonal resp. unitary groups (e.g. this phrase is used in Bott's classical paper on Bott periodicity); these groups are not stable homotopy groups in the sense of stable homotopy theory.
With regard to characteristic classes, the Stiefel-Whitney and Pontryagin classes are all stable in the sense that they are all pulled back from characteristic classes on $BO$, so  they can only see the stable equivalence class of a vector bundle, and similarly the Chern classes are all pulled back from characteristic classes on $BU$.  The Euler class of an oriented real vector bundle is unstable, so it can tell you, for example, that the tangent bundle to an even-dimensional sphere $S^{2n}$ is nontrivial even though it is stably trivial. 
Stable homotopy classes of maps $M \to BO$ resp. $M \to BU$ are the wrong thing to ask for. If I'm not mistaken, this is equivalent to asking for homotopy classes of maps between the suspension spectra of $M$ and $BO$ resp. $BU$, but taking suspension spectra destroys information about $BO$: it is already an infinite loop space (by Bott periodicity!), hence a connective spectrum, and so maps $M \to BO$ resp. $M \to BU$ already factor through the suspension spectrum of $M$. That is, the set of homotopy classes of maps $M \to BO$ resp. $M \to BU$ is already an invariant of the stable homotopy type of $M$. I think when $M$ is connected it can be identified with reduced real K-theory $\widetilde{KO}^0(M)$ resp. reduced complex K-theory $\widetilde{KU}^0(M)$. 
A: The notion of "vector bundles up to stable equivalence" isn't defined for a general topological group $G$.  Stable equivalence for vector bundles is defined in terms of taking direct sums with trivial bundles, but there is no analogous notion for a general topological group.  So one side of the question is ill-defined.  
