The aim of this paper is to obtain new fine properties of entropy solutions of nonlinear scalar conservation laws. For this purpose, we study some ''fractional $BV$ spaces'' denoted $BV^s$, for $0 < s \leq 1$, introduced by Love and Young in 1937. The $BV^s(\mathbb{R})$ spaces are very closed to the critical Sobolev space $W^{s,1/s}(\mathbb{R})$. We investigate these spaces in relation with one-dimensional scalar conservation laws. $BV^s$ spaces allow to work with less regular functions than BV functions and appear to be more natural in this context. We obtain a stability result for entropy solutions with $BV^s$ initial data. Furthermore, for the first time we get the maximal $W^{s,p}$ smoothing effect conjectured by P.-L. Lions, B. Perthame and E. Tadmor for all nonlinear degenerate convex fluxes.