This is more a remark than an answer.


It is perhaps worth noting that in the related classification of foliations on surfaces by [McQuillan][1], [Brunella][2], and [Mendes][3] abundance does not hold. The so called Hilbert Modular foliations are examples of foliations with nef canonical bundle but with Kodaira-Iitaka
dimension negative. These turn out to be the only examples, and the proof of this fact is the harder part of the classification. 


  [1]: https://mathscinet.ams.org/mathscinet-getitem?mr=2435846
  [2]: https://mathscinet.ams.org/mathscinet-getitem?mr=1948251
  [3]: http://www.springerlink.com/content/q923t4141q266132/