Though it's relatively clear that the characteristic classes do not characterise a vector bundle (and after looking through some books) I could not find an example of a vector bundle which is not stably trivial but whose characteristic classes (those which may be defined\*) are all trivial. Could someone be so kind as to point out a reference for this?


\*an example with trivial Stiefel-Whitney class would already be nice.



Non-stably trivial bundle with trivial characteristic classes
You will find an answer to your question in Hatcher's book project "Vector bundles an K-theory" (p. 75-76) (available on his homepage). 
Using the fact that $\pi\_8(O(10))=\mathbb Z\_2$, you can build a non- stably trivial vector bundle over the sphere $S^9$ (using the clutching function associated to the non-trivial homotopy class $S^8\rightarrow O(10)$ in $\pi\_8(O(10))$). This vector bundle has all his Stiefel-Whitney and Pontryagin classes equal to zero. 


The vanishing of $w\_9$ follows from Wu's formula $w\_9=w\_1w\_8+Sq^1(w\_8)$. 


