What's the best method to extract vocals from music?
The example sounds like pitch tracking combined with comb filtering. Using method like this could be implemented by "drawing" as well, as in SPEAR analysis and re-synthesis program.

If the vocal is in the middle and other instruments are not, then you can also try center channel isolation:

http://www.virtualdub.org/blog/pivot/entry.php?id=102

The methods however, sound horrible in addition to being inaccurate and full of artifacts. Better solution to the problem would be a program that could effectively guess which part of the signal belongs to the voice and which doesn't. Much more difficult for sure though.