This topic from Long and Short Reviews was a tough one. I can think of plenty of villains I liked purely because of who played them in the film, but I still wouldn't say I would root for them over the protagonist. Even if they were played by Alan Rickman, who was always amazing at playing villains.
Then I thought of "Mansfield Park" by Jane Austen.
The heroine Fanny Price, a painfully good and moralistic girl, is pursued by rakish Henry Crawford while being secretly in love with her equally prim and proper cousin Edmund. We'll bypass the whole cousin-marriage thing for a moment since apparently that was fine at the time. Henry Crawford is charming, entertaining and - most importantly - is very much in love with Fanny Price, despite having misbehaved with women in the past. Edmund, on the other hand, is hopelessly in love with Henry's sister and completely unaware of Fanny's feelings, while also disapproving of basically everything interesting in life. I find it impossible to prefer Edmund over Henry.
Of course, Fanny is an insufferable character, so she probably deserves to end up with her equally annoying cousin. But if I had been her, I would have run off with Henry Crawford any day of the week.
So which villains would you root for?