Each State also has its own Supreme Court to oversee the state court system. However, its name is not always “Supreme Court.” In New York, oddly enough, the “Supreme Court” is the lowest level, where trials are held. The highest appellate court is the “New York Court of Appeals.” | Each State also has its own Supreme Court to oversee the state court system. However, its name is not always “Supreme Court.” In New York, oddly enough, the “Supreme Court” is the lowest level, where trials are held. The highest appellate court is the “New York Court of Appeals.” |