Every (closed, connected, oriented) three-manifold contains a fibred knot.  That is, every three-manifold has an open book decomposition with non-empty, connected binding and connected, oriented, compact page. This is due to Winkelnkemper or Gonzalez-Acuna - see page 625 of this [book][1].  

**Edited** 

So, $M = \Sigma \times S^1$ has the "obvious" open book decomposition (if we allow empty bindings). By the above $M$ also has (infinitely many) open book decompositions with non-empty bindings.  I very much doubt that there is a preferred one among the latter.

  [1]: https://www.maths.ed.ac.uk/~v1ranick/books/knot.pdf