  We prove that if $u$ is the entropy solution to a scalar conservation law in
one space dimension, then the entropy dissipation is a measure concentrated on
countably many Lipschitz curves. This result is a consequence of a detailed
analysis of the structure of the characteristics. \\ In particular the
characteristic curves are segments outside a countably 1-rectifiable set and
the left and right traces of the solution exist in a $C^0$-sense up to the
degeneracy due to the segments where $f"=0$.
  We prove also that the initial data is taken in a suitably strong sense and
we give some counterexamples which show that these results are sharp.
