The total space of a fiber bundle over a sphere that isn't a circle has zero signature by a <a href="http://hirzebruch.mpim-bonn.mpg.de/163/1/15_On%20the%20index%20of%20a%20fibered%20manifold.pdf"> result </a>of 
Chern, Hirzebruch, and Serre that the signature is multiplicative when the fundamental group of the base acts trivially on the cohomology of the fiber.