Migraine headaches. What kinds of wine cause them, and which ones don't?

Depends on person. While red wine is a more common migraine trigger than white (possibly due to the Amino Acid tyramine), there is significant person-to-person variability. Keeping track of your personal triggers is the best approach overall.
Red wine. People who have migraines have different triggers. One common trigger for migraine headaches is red wine; although their are many people who are predisposed to migraine headaches for whom red wine is not a trigger. Each migraine sufferer must learn their own triggers. For more information, see: http://understandingmigraine.blogspot.com/
Any kind. Red wines are not an infrequent cause but it is very personal and there is no general rule about any of the numerous trigger factors in migraine.