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