Given a positive integer $n$, there is a well known free action of $\mathbb T^1$ on $\mathbb S^{2n-1}$ due to Hopf, which makes $\mathbb S^{2n-1}$ a fibre bundle with the fibre $\mathbb T^1$. Moreover, for a few triples of positive integers $k,l,m$, $\mathbb S^l$ is a fibre bundle over $\mathbb S^m$ with the fibre $\mathbb S^k$.

Could you please present other, maybe less known, examples of structures of a fibre bundle on spheres $\mathbb S^n$ for various (classes of) $n$?