Theorem of Takeuchi (in Free Hopf algebras generated by coalgebras, 1971) asserts that free Hopf algebra $H(C)$ over a coalgebra $C$ has injective antipode, and it is bijective precisely (at least over alg. closed field) when $C$ is pointed.

On the other hand, in a paper Faithful flatness over Hopf subalgebras - counterexamples, 2000 P. Schauenburg constructs a Hopf algebra with surjective, but non-injective antipode. Example is constructed as follows. There's a functor of "free Hopf-with-bijective-antipode" functor from coalgebras to Hopf algebras; one can find a biideal in such free Hopf algebra over matrix coalgebra that is stable under antipode, but not under its inverse, and quotient will be the example. Paper can be found on author's homepage http://schauenburg.perso.math.cnrs.fr/personnelle.html

Theorem of Takeuchi (in Free Hopf algebras generated by coalgebras, 1971) asserts that free Hopf algebra $H(C)$ over a coalgebra $C$ has injective antipode, and it is bijective precisely (at least over alg. closed field) when $C$ is pointed.

On the other hand, in a paper Faithful flatness over Hopf subalgebras - counterexamples, 2000 P. Schauenburg constructs a Hopf algebra with surjective, but non-injective antipode. Example is constructed as follows. There's a "free Hopf-with-bijective-antipode" functor from coalgebras to Hopf algebras; one can find a biideal in such free-with-bijective-antipode Hopf algebra over matrix coalgebra $M_4(k)$ that is stable under antipode, but not under its inverse, and quotient will be the example. Paper can be found on author's homepage http://schauenburg.perso.math.cnrs.fr/personnelle.html