Let $f: M \to M$ be a $C^{1+\alpha}$ diffeomorphism preserving an invariant probability measure $\mu$. Assume either that $M$ is compact, or in the noncompact Riemannian case that the Oseledets integrability conditions \begin{align*} \log^+\|Df\|_{\mathrm{op}},\ \log^+\|Df^{-1}\|_{\mathrm{op}} \in L^1(M,\mu) \end{align*} hold in local Riemannian norms. At $\mu$-a.e. point whose Lyapunov exponents are nonzero, there are local stable and unstable manifolds tangent to the Oseledets stable and unstable subspaces.