I know this book which is not currently in our library though. But I'm not sure kind of examples exist there, do they?

Dear Laurent Moret-Bailly: Thank you very much for this nice example! I was also wondering whether we can move one more step to ask when the linear system induces a birational morphism, whether the counterexample still exists.

To S\'andor: Thanks for your remark! Yes, my question is whether such examples exist.

To Garcia: What I said is all for the general log pair case. I reedit my answer because I realize that Keel only show in dimension 3, flipping and divisorial contractions exist as EWM. In other words, so far we only know the target space for these contracts exists as an algebraic space.

I see. You are right. We also need to consider the curves of discrepancy 0 which doesn't necessarily appear in the minimal resolution of the surface itself.

Dear Jason: By 'MMP', I mean precisely the things we need for running the minimal model program (in the surface case, it just means the contraction exists.). With further digging on the literature, it might be also true that the abundance is also known for the surfaces pairs in char p. On the other hand, in the paper "Strong Rational Connectedness of Surfaces", there is an example due to Koll'ar giving a del Pezzo surface with only A_1 singularities, whose smooth locus doesn't contain free curves. Of course it still may be rationally connected, which I didn't really check.