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]: http://ams.impa.br/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=ALLF&pg7=ALLF&pg8=ET&r=1&review_format=html&s4=mcquillan&s5=canonical&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq
  [2]: http://ams.impa.br/mathscinet/search/publdoc.html?pg1=MR&s1=1948251&loc=fromreflist
  [3]: http://www.springerlink.com/content/q923t4141q266132/