  We consider singular foliations of codimension one on 3-manifolds, in the
sense defined by A. Haefliger as being Gamma_1-structures. We prove that under
the obvious linear embedding condition, they are Gamma_1-homotopic to a regular
foliation carried by an open book or a twisted open book. The latter concept is
introduced for this aim. Our result holds true in every regularity C^r, r at
least 1. In particular, in dimension 3, this gives a very simple proof of
Thurston's 1976 regularization theorem without using Mather's homology
equivalence.
