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

On the structure of $L^\infty$-entropy solutions to scalar conservation laws in one-space dimension