For any ring $R$, a non-zero module $S$ is stably free if $S\oplus R^a$ is free ($a\geq 1$). This may be an overly vague question, but I am wondering in what contexts do stably free modules arise in algebraic geometry?

I know Serre asked whether projective modules (that turn out to be stably free) over $k[x_1,\ldots, x_n]$ ($k$ field) are free. This, of course, corresponds to asking whether vector bundles over the affine $n$-space $\mathbb{A}^n_k$ are trivial. However, this question was asked back in the 50s (eventually being answered that all projective modules are free!) and I am wondering what role stably free modules now play (or can play) in algebraic geometry.