  Nonparametric estimation for semilinear SPDEs, namely stochastic
reaction-diffusion equations in one space dimension, is studied. We consider
observations of the solution field on a discrete grid in time and space with
infill asymptotics in both coordinates. Firstly, based on a precise analysis of
the H\"older regularity of the solution process and its nonlinear component, we
show that the asymptotic properties of diffusivity and volatility estimators
derived from realized quadratic variations in the linear setup generalize to
the semilinear SPDE. In particular, we obtain a rate-optimal joint estimator of
the two parameters. Secondly, we derive a nonparametric estimator for the
reaction function specifying the underlying equation. The estimate is chosen
from a finite-dimensional function space based on a simple least squares
criterion. Oracle inequalities with respect to both the empirical and usual
$L^2$-risk provide conditions for the estimator to achieve the usual
nonparametric rate of convergence. Adaptivity is provided via model selection.
