At least twice in the last few weeks I've come across someone claiming that the U.S. Supreme Court affirmed the one-drop rule in 1986. I was surprised, because shortly before the first time I saw this claim I'd come across someone else saying that the 1967 case Loving v. Virginia, which is best known for overturning Virginia's ban on interracial marriage, also declared the one-drop rule unconstitutional. So I eventually started looking into both claims. It turns out that the first is false, and the second is true. That is, the Supreme Court did overturn one-drop-rule style racial classification laws in 1967, and they did not affirm a one-drop-rule law in 1986.
What Chief Justice Earl Warren's opinion in Loving actually says in the main text is that racial classifications need to be subjected to the most rigid scrutiny, especially if they form the basis of some impact in a criminal proceeding. But this isn't a new judgment. It's a quotation of a previous decision. And it's not clear what the most rigid scruntiny is supposed to be or how it would apply to one-drop rule laws, and he never applies it to such laws. But he points out that the basis of the racial classifications used in the Virginia law were instituted specifically to preserve the conception of white purity advocated by the invidious discrimination of 1924 Virginia that was of a piece with the kind of segregation at odds with the Equal Protection clause of the 14th Amendment, and that can't stand up to the most rigid scrutiny.
It's not quite clear, however, until you get to footnote 11, which says that the racial-classification system of Virgina is "repugnant to the Fourteenth Amendment" (and therefore presumably unconstitutional, although he never explicitly says they're overturning that law too). Since this is the reasoning for the overturning of the interracial-marriage ban, and not some aside on a topic not necessary for guiding the current case, I think it does count as overturning one-drop rule laws, at least any justified on the basis of white supremacy or purity (as I'm sure all actual one-drop rule laws were). But I now understand how it can do that in a way that I didn't really notice before. The real work is done in a footnote.
But the first claim is simply false. What happened in 1985 was a case involving a Louisiana woman who had thought of herself as white all her life who then discovered that her birth certificate listed her parents as colored. Louisiana law, until 1983, had a 1/32 one-drop rule, which counted someone as colored for having one black ancestor out of 32 great-great-great grandparents. Her parents were classified as colored by that law. She herself actually didn't count as black by that law, since it was her great-great-great-great grandmother who was black. But her birth certificate listed her as colored because her parents were listed as colored on theirs. So it wasn't the one-drop rule law that led her to be classified as black on her birth certificate. It was the cultural practice among doctors and midwives of transferring the racial-classification of the parents to the child when both parents had the same classification. Her parents had never objected to their classifications, and corrections to birth certificates apparently had to come from the person whose birth certificate it is issuing a complaint and request for correction.
So the state court concluded that there was no legal justification for forcing the birth certificate office to issue corrected birth certificates. They then said that the repealed 1/32 one-drop rule law was not relevant, because midwives and doctors aren't subject to the prohibition on government employees' violation of the 14th Amendment, since they're not government employees. Finally, they said the one-drop rule laws involved with this did, by their judgment, violate the Constitution, but they were bound by Louisiana Supreme Court precedent on that question. None of their analysis depended on any stance on the one-drop rule law, which was no longer on the books at this time anyway and thus could not be overturned by a court in any direct way. The case apparently got appealed to the Supreme Court in 1986, and they opted not to hear it, but it seems crazy to me to take that as a sign that they would affirm a one-drop-rule law.