We show that most homogeneous Anosov actions of higher rank Abelian groups are locally $C^\infty$-rigid (up to an automorphism). This result is the main part in the proof of local $C^\infty$-rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nilmanifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper “onstationary” generalization of the classical theory of normal forms for local contractions.