  This paper deals with the optimal regularity for entropy solutions of
conservation laws. For this purpose, we use two key ingredients: (a) fine
structure of entropy solutions and (b) fractional $BV$ spaces. We show that
optimality of the regularizing effect for the initial value problem from
$L^\infty$ to fractional Sobolev space and fractional $BV$ spaces is valid for
all time. Previously, such optimality was proven only for a finite time, before
the nonlinear interaction of waves. Here for some well-chosen examples, the
sharp regularity is obtained after the interaction of waves. Moreover , we
prove sharp smoothing in $BV^s$ for a convex scalar conservation law with a
linear source term. Next, we provide an upper bound of the maximal smoothing
effect for nonlinear scalar multi-dimensional conservation laws and some
hyperbolic systems in one or multi-dimension.
