New York's naming runs backward
Every county in New York has a branch of the Supreme Court, which is the state's main trial-level court for civil cases. Many significant civil lawsuits, including contract disputes, personal injury claims, and divorces, are filed there. Appeals from it generally go to the Appellate Division, and the state's highest court is the Court of Appeals. That arrangement is the reverse of most states and of the federal system, so people from elsewhere, and many New Yorkers too, find it confusing. A Supreme Court lawsuit in New York is therefore usually the start of a case, not its last stage.
If you have been served
A summons from New York Supreme Court comes with a deadline to respond, and the time allowed depends on how the papers were delivered. A default judgment can follow if that deadline passes, and undoing one takes far more effort than filing an answer on time. Keep the envelope, note the date and manner of delivery, and gather the contracts, emails, and records related to the claim. An insurer that may cover the claim should hear about it promptly, since many policies make timely notice a condition. Even a claim you consider baseless needs a response, and arguing it out directly with the other side tends to complicate matters rather than end them.
The United States Supreme Court
If you meant the nation's highest court, that is a different matter. The U.S. Supreme Court hears relatively few cases, mostly on discretionary review after other courts have ruled, and the request for review, a petition for certiorari, must be filed within a strict period after the lower court's decision. Cases rarely start there; they reach it after moving through other courts, and most petitions are denied. In a first conversation we confirm which court is involved, what stage the case has reached, and what deadline applies, and then decide whether to answer, move to dismiss, or prepare an appeal.
