AP Style rules hold that you should capitalize state supreme courts with the state name and also without the state name when the context makes it unnecessary. For example,
- the New Jersey Supreme Court
- the state Supreme Court
- the Supreme Court
If a court with this name is not a state’s highest tribunal, the fact should be noted. In New York, for example, the Supreme Court is a trial court. Appeals are directed to the Appellate Division of the Supreme Court. The state’s highest court is the Court of Appeals.