A smooth non-stably trivial smooth vector bundle This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to th...--prophetes.ai

A smooth non-stably trivial smooth vector bundle This may well be just a look-up, but do you have an example of a non-stably trivial smooth vector bundle? If it has a presentation as the vector bundle associated to the representation of some (classical) Lie group, I'd be very happy.

The Möbius bundle over $S^1$ has this property.