"Black magic" is a really large umbrella for a wide range of claims, but the very definition of any "magic" is that it is somehow supernatural. Naturally (!), no compelling evidence has ever been put forth for any supernatural phenomenon. Once a phenomenon is observable, reproducible, and testable, and shown to exist, it may turn out to violate our current understanding of the natural laws, but if the phenomenon is for real, we shall have to adjust our views to accommodate this, and the phenomenon shall cease to be considered supernatural. "Magic" will never be shown to exist.
All rigorously tested supernatural claims have turned out to be fake, and the great number of untested supernatural claims can most likely be largely attributed to the fact that the practitioners are aware of their fraud, and reluctant to be exposed; see the million dollar challenge in Regebro's answer.
More specifically, in regions of India there is a somewhat widespread belief in tantra. While few practitioners would lend themselves to scientific studies for reasons explained above, there has been one notable appearance where Indias allegedly most powerful tantrik was challenged in live TV in front of an audience of millions of people, to kill a person with the aid of black magic alone.
Rationalist International has a good writeup on the story. In short, the tantrik claimed he could kill any person he wanted within three minutes, but did not manage to inflict any sort of damage during what went on for more than three hours. Rationalist International concludes:
Tantra power had miserably failed. Tantriks are creating such a scaring atmosphere that even people, who know that black magic has no base, can just break down out of fear, commented a scientist during the program. It needs enormous courage and confidence to challenge them by actually putting one’s life at risk, he said. By doing so, Sanal Edamaruku has broken the spell, and has taken away much of the fear of those who witnessed his triumph.
