Almost all cases in the Supreme Court are argued orally. Rarely does the Supreme Court of the United States decide a case without any oral argument.
Occasionally it will decide a case "per curiam" (Latin for "by the Court"). This does not mean that all the Justices agree; it just means that no individual judge is listed as the author of the opinion. Judges may, and sometimes do, dissent by name from per curiam decisions.
The U.S. Supreme Court can also change its own mind about previously granting a Petition for Certiorari and, either before or after oral argument, dismiss the Petition as "improvidently granted" in legalese. This makes the lower court's decision final. It is usually done because on closer examination the case does not clearly present the issue(s) the Court wanted to decide.