I've been seeing some headlines referring to decisions of the New York Supreme Court (see, e.g., Fox and CNN), but that's likely to be confusing to most readers: In New York,
- "Supreme Court" refers to the trial court, though in most other states it's the name of the state highest court, and
- "Court of Appeals" refers to the highest court, though in most other states it's the name of the state intermediate court.
So when you hear that "the New York Supreme Court has struck down a law …," that just means that one trial court judge has made that decision, which would often get appealed (and perhaps reversed) within the state court system. I therefore try to use "New York trial court" or "New York high court" instead, as the case might be. (Technically, New York has some other trial courts as well, but I think "trial court" for "Supreme Court" is likely to be the least confusing option.)
