Is it true that any $R^n$-bundle over a space (say a simplicial space) of dimension $k<n$ is trivial? It seems to me any $R^n$-bundle ,for $n>1$, over $S^1$ is trivial. But I cannot figure out if this is true in general.
Also can you give you an example of a stably trivial vector bundle? i.e. a vector bundle $p:E\to B$ such that $E\oplus\epsilon^k$ ($\epsilon^k$ being a trivial $R^k$-bundle over $B$) is trivial. Again I think if we take $p:E\to S^1$ be the mobius bundle and $\epsilon^1$ the trivial bundle over $S^1$, then their sum is a trivial $R^2$-bundle. Can you provide other examples?
For the first question, take a bundle $\xi \to B$ with $w(\xi)\not = 1$, and note that $w(\xi\oplus \theta^n) = w(\xi) w(\theta^n) = w(\xi)\not = 1$ for all $n$, where $\theta^n \to B$ is the trivial bundle.
For the second question, take $TS^n \to S^n$. The normal bundle of $\nu \to S^n$ in $\mathbb{R}^{n+1}$ is trivial, since it admits the nowhere-vanishing section $s(p) = p$; and by definition $TS^n \oplus \nu = \theta^{n+1}$.