What kind of cases does the Supreme Court hear?
Almost all of the cases that the Supreme Court hears are cases that are on appeal. The Supreme Court does have original jurisdiction over a very few cases, but these are quite rare. This means that the Supreme Court is almost always hearing cases where only matters of law are at issue (rather than matters of fact). The Supreme Court is simply, in those cases, trying to decide if the law (whether statute law or the Constitution) has been correctly applied.
Cases heard by the Supreme Court generally involve very important and difficult issues of law. Cases that are not important, or where the law is obvious, do not make it all the way up the ladder to the Supreme Court.
So, the cases the Court hears are those that involve important and difficult questions of law. It hears those cases either after they have come up through the federal court system or after they have been decided by the supreme court of a state.
In addition to what has been stated, the SC can also hear a case that was at a state Appeals court level when that state's SC refused to hear it.
Here is one such 4th AM case where the AZ SC denied review. Since it dealt with the federal constitution, of course, it contained the "federal question" necessary for review.
At times, when a decision, even though when it rests primarily on state law, but contains references to the federal constitution/laws enough to tangle it, the US SC can grant review.
In rare cases, such as Marshall v. Marshall, the Anna Nicole Smith case, there is a rare probate/domestic relations exception, even though it is soley state law.
One time in the Court's history, it held a criminal trial themselves, NOT an appeal. This was the case of Sheriff Joseph Shipp, etc. al., a Contempt of Court case.
The USSC does NOT issue what are known as "Advisory Opinions", or a advisory on law, without an appeal.
At one time the court was mandated to hear all appeals before it, this changed, if I am not mistaken, with the Judiciary Act of 1925. They can pick and choose which cases to hear and it takes 4 out of the 9 Justices to grant an Appeal, known as Certiorari.
A lower court decision which rests SOLEY on state law, no federal element present, can not be heard by the US SC.
They can hear Petitions for a STAY of execution from a court, this is submitted to the Justice who has the Alottment for that state. If granted, the execution can not go forward until the SC itself or the Justice him or her self lifts it.