  We present results from a lattice QCD study of nucleon matrix elements at
vanishing momentum transfer for local and twist-2 isovector operator
insertions. Computations are performed on gauge ensembles with
non-perturbatively improved $N_f=2+1$ Wilson fermions, covering four values of
the lattice spacing and pion masses down to $M_\pi\approx200$MeV. Several
source-sink separations (typically ~1.0fm to ~1.5fm) allow us to assess
excited-state contamination. Results on individual ensembles are obtained from
simultaneous two-state fits across all observables and all available
source-sink separations with the energy gap as a common fit parameter.
Renormalization has been performed non-perturbatively using the
Rome-Southampton method for all but the finest lattice spacing for which an
extrapolation has been used. Physical results are quoted in the $\overline{MS}$
scheme at a scale of $\mu=2$GeV and are obtained from a combined chiral,
continuum and finite-size extrapolation. For the nucleon isovector axial,
scalar and tensor charges we find physical values of
$g_A^{u-d}=1.242(25)_\text{stat}(\genfrac{}{}{0pt}{2}{+00}{-31})_\text{sys}$,
$g_S^{u-d}=1.13(11)_\text{stat}(\genfrac{}{}{0pt}{2}{+07}{-06})_\text{sys}$ and
$g_T^{u-d}=0.965(38)_\text{stat}(\genfrac{}{}{0pt}{2}{+13}{-41})_\text{sys}$,
respectively, where individual systematic errors in each direction from the
chiral, continuum and finite-size extrapolation have been added in quadrature.
Our final results for the isovector average quark momentum fraction and the
isovector helicity and transversity moments are given by $\langle
x\rangle_{u-d}=0.180(25)_\text{stat}(\genfrac{}{}{0pt}{2}{+14}{-06})_\text{sys}$,
$\langle x\rangle_{\Delta u-\Delta
d}=0.221(25)_\text{stat}(\genfrac{}{}{0pt}{2}{+10}{-00})_\text{sys}$ and
$\langle x\rangle_{\delta u-\delta
d}=0.212(32)_\text{stat}(\genfrac{}{}{0pt}{2}{+20}{-10})_\text{sys}$,
respectively.
