  Higher Sobolev and H\"older regularity is studied for local weak solutions of
the fractional $p$-Laplace equation of order $s$ in the case $p\ge 2$.
Depending on the regime considered, i.e. $$0<s\le\tfrac{p-2}{p}\quad \text{or}
\quad\tfrac{p-2}{p}<s<1,$$ precise local estimates are proven. The relevant
estimates are stable if the fractional order $s$ reaches $1$; the known Sobolev
regularity estimates for the local $p$-Laplace are recovered. The case $p=2$
reproduces the almost $W^{1+s,2}_{\rm loc}$-regularity for the fractional
Laplace equation of any order $s\in(0,1)$.
