In the cosmological scenario in $f(T)$ gravity, we find analytical solutions for an isotropic and homogeneous universe containing a dust fluid and radiation and for an empty anisotropic Bianchi I universe. The method that we apply is that of movable singularities of differential equations. For the isotropic universe, the solutions are expressed in terms of a Laurent expansion, while for the anisotropic universe we find a family of exact Kasner-like solutions in vacuum. Finally, we discuss when a nonlinear $f(T)$-gravity theory provides solutions for the teleparallel equivalence of general relativity and derive conditions for exact solutions of general relativity to solve the field equations of an $f(T)$ theory.