We shall consider sections of a complex elliptic scheme ε over an affine base curve B, and study the points of B where the section takes a torsion value. In particular, we shall relate the distribution in B of these points with the canonical height of the section, proving an integral formula involving a measure on B coming from the so-called Betti map of the section. We shall show that this measure is the same one which appears in dynamical issues related to the section. This analysis will also involve the multiplicity with which a torsion value is attained, which is an independent problem. We shall prove finiteness theorems for the points where the multiplicity is higher than expected. Such multiplicity has also a relation with Diophantine Approximation and quasi-integral points on ε (over the affine ring of B), and in Sections 5 and 6 of the paper we shall exploit this viewpoint, proving an effective result in the spirit of Siegel's theorem on integral points.
