The Thom space of the tangent bundle of $S^n$ by the Thom isomorphism has reduced cohomology $\mathbb{Z}$ in dimensions $n$ and $2n$. It is also the case that it suspends to $S^{2n+1} \vee S^{n+1}$ since adding a trivial bundle makes it trivial, and the Thom space of a trivial bundle is the repeated suspension of the space with a disjoint basepoint.
Is this just a stable equivalence or is it always true that the Thom space of the sphere is a wedge of spheres? Perhaps this can be reduced to a question about Hopf invariants?