Does SCOTUS have jurisdiction over all federal cases, or must it go through "normal channels?"

Question

In the United States, what are the required steps before the Supreme Court can hear a case?

Obviously, we are assuming the court would have jurisdiction (e.g., it would need to be a federal - not a state matter, and there would have to be an actual case, not just a desire), but once a legitimate suit has been brought in a federal court, are there any requirements for what can be heard?

In a great answer, I'd like to know what is common and what is required. For example, I know that a case is typically heard in a lower court, an appellate court will decide any matters of process, and only if the Supreme Court desires to hear a case will it come onto their docket - but I'm also interested in knowing which steps are actually required.

Answer 1

As described in Wikipedia, in theory the Court can function both as apellate court:

By petition for an "extraordinary writ" such as mandamus, prohibition, or habeas corpus.

and as original jurisdiction court:

Certain cases that have not been considered by a lower court may be heard by the Supreme Court in the first instance under what is termed original jurisdiction. The Supreme Court's authority in this respect is also derived from Article III of the Constitution, which states that the Supreme Court shall have original jurisdiction "in all cases affecting ambassadors, other public ministers and consuls, and those in which a state shall be party."

So the strict succession of lower court decision, then appellate court decision, and only then the Supreme Court can accept the case - is not required. Even though it is not how the things usually happen, the rules of the Supreme Court explicitly describe the procedure for filing a direct petition as Rule 17.