Pick a Yak

It may seem random, but actually the Yaks will be found to be the following (in this order):

1. 
   
2. Mean
3. 
   
4. Smelly
5. 
   
6. Angry
7. 
   
8. Stabby
9. 
   
10. Mean Angry
11. 
    
12. Stabby Smelly
13. 
    
14. Angry Stabby
15. 
    
16. Mean Smelly
17. 
    
18. Mean, Smelly, Angry, Stabby, Very Bad Townlong Yak
19. 
    
20. Suitable Yak
21. 
    

So, basically... kill 9 yaks, the 10th will be the best.

Does this remind anyone else of the children's book: "Alexander and the Terrible, Horrible, No Good, Very Bad Day?" :)

Really surprised this information isn't here, but in order for this quest to appear you must complete the two quests:

The Motives of the Mantid

And

Set the Mantid Back

Both from Tai Ho at Shando-Pan Garrison (49,72).

He'll give you a quest afterwards called The Wisdom of Niuzao. Just fly over to the turn in (you don't actually have to turn it in), and accept the quest from Bluesaddle (standing right next to the turn in) called Bad Yak. Do that real quick and he'll appear at 35,57 with this quest.

You can run away to reset the yak, which saves on a lot of time spent killing - it'll still move on to the next step in the chain even if you don't actually kill it. (You can also just keep lassoing the same yak this way!)