How does a case reach the US Supreme Court?

Answer

There's only one way. It must wend its way through the federal or state court system - even then, it must be selected to be heard (usually having to do something with US Constitutional issues).

Answer

Assuming you're referring to cases heard under the Supreme Court's appellate jurisdiction, there is a second way the case may come before the Court: through certification of a question from one of the Circuit Courts.

The US Court of Appeals Circuit Courts may petition the US Supreme Court to answer a constitutional question (which then becomes binding on the lower court). When the Circuit Court certifies a question, the Supreme Court has the option of requesting the entire case be sent to them for review.

While this process is still listed as one official method for getting a case before the Supreme Court, the justices dislike certification and haven't granted one since 1984.