Sphere bundles and bundles over spheres are everywhere and are excellent things to get one's hands dirty with.

(1a) But when can we have a bundle $S^n \to S^m \to B?$ It seems like requiring the total space of a sphere bundle to be a sphere is pretty restrictive.

(1b) Does the answer to (1a) depend on a choice of category (PL, TOP, etc)?

There's an $S^1-$ bundle over $CP^1$ with total space $S^3$, but that's the only example I can find.

(2) Do people know examples other than the one above?