Corollary 9 in these notes by Ralph Cohen https://cmsa.fas.harvard.edu/wp-content/uploads/2022/03/immersions-revised2.pdf
has grabbed my attention.

>I do not undestand how to show that if we have a rank $k$ bundle which is *stably* isomorphic to the stable normal bundle then there is a virtual normal bundle of rank $k$.

This seems to boil down to proving the following thing:


Let $M^m$ a smooth manifold  and suppose we are given $f:M^m\to BO(k)$. 
Let  $n>k$ and let   $g: M^m\to BO(n)$ be  such that 
$$TM\oplus g^*(EO(n)) \simeq \varepsilon^{n+m}$$
in other words, g is the classifying map for an virtual normal bundle of rank $n$ ($\varepsilon$ is the trivial bundle).

Suppose that $g$ is homotopic to $i\circ f$ where $i:BO(k)\to BO(n)$ is the obvious inclusion.
Then we would like to show that
 $$TM\oplus f^*(EO(k)) \simeq \varepsilon^{k+m}.$$

However the only thing I can conclude from the homotopy $g\simeq i\circ f$ is that 
$$TM\oplus f^*(EO(k))\oplus \varepsilon^{n-k} \simeq \varepsilon^{n+m}$$, which is not enough in general.