  We study weak-field solutions having spherical symmetry in $f(T)$ gravity; to
this end, we solve the field equations for a non diagonal tetrad, starting from
Lagrangian in the form $f(T)=T+\alpha T^{n}$, where $\alpha$ is a small
constant, parameterizing the departure of the theory from GR. We show that the
classical spherically symmetric solutions of GR, i.e. the Schwarzschild and
Schwarzschild-de Sitter solutions, are perturbed by terms in the form $\propto
r^{2-2n}$ and discuss the impact of these perturbations in observational tests.
