This paper deals with a sharp smoothing effect for entropy solutions of one-dimensional scalar conservation laws with a degenerate convex flux. We briefly explain why degenerate fluxes are related with the optimal smoothing effect conjectured by Lions, Perthame, Tadmor for entropy solutions of multidimensional conservation laws. It turns out that generalized spaces of bounded variation $BV_Φ$ are particularly suitable -better than Sobolev spaces- to quantify the regularizing effect and to obtain traces as in BV. The function $Φ$ in question is linked to the degeneracy of the flux. Up to the present, the Lax-Oleĭnik formula has provided optimal results for an uniformly convex flux. In this paper we first need to validate this formula for a more general flux, i.e. a $C^1$ strictly convex flux. We also give a complete proof that for a nonlinear degenerate convex flux the Lax-Oleĭnik formula provides the unique entropy solution, namely the Kružkov solution.