Here is a result that I randomly stumbled upon. Let us call a space W-trivial if every real vector bundle over it has trivial total Stiefel-Whitney class. Atiyah and Hirzebruch now proved in the 60s: If Y is a finite CW-complex, then its 9-fold suspension is W-trivial.
Why did they prove such a result? They wanted to give yet another proof that the only parallelizable spheres are the ones in dimensions 1, 3 and 7 (at that time four other proof already existed). The result that 9-fold suspensions are W-trivial implies, in combination with an argument by Milnor, that spheres of dimensions 9 and higher are not parallelizable.
How did they prove it? They used Bott periodicity for KO-theory. Note first that the computation of the total Stiefel-Whitney class of a real vector bundle \(\eta\) over \(S^9 \wedge Y\) factors through \(KO(S^9 \wedge Y)\). By Bott periodicity we know that this KO-element comes from an element in \(\widetilde{KO}(S^1 \wedge Y)\); concretely, \(\eta\, – k = \beta(x)\), where \(\beta\) is the Bott periodicity isomorphism and k the rank of \(\eta\). The main technical work of Atiyah and Hirzebruch is now to derive a formula for the total Stiefel-Whitney class \(\omega(\beta(x))\) of \(\beta(x)\) in terms of the Stiefel-Whitney classes of \(x\). This formula shows that \(\omega(\beta(x))\) mainly consists of cup products of the Stiefel-Whitney classes of \(x\). But we know that the cohomology ring of \(S^1 \wedge Y\) is trivial (this is true for any suspension) and hence they can conclude that \(\omega(\beta(x))\) is trivial.