  We obtain several new regularity results for solutions of scalar conservation
laws satisfying the genuine nonlinearity condition. We prove that the solutions
are continuous outside of the jump set, which is codimension one rectifiable.
We show that the entropy dissipation vanishes away from the closure of the jump
set. We prove that the solution decays algebraically in $L^\infty$ as $t \to
\infty$ and we compute the presumably optimal decay rate. All these results are
based on a local oscillation estimate which is obtained properly adapting some
ideas of De Giorgi from the context of elliptic equations.
