Can state court cases be appealed beyond the state supreme court?

It depends on the circumstance.

If the issue in question (the issue that caused the appeal) is an error regarding state law, then the state supreme court would be the final appeal.

If the issue in question is an error regarding federal law, then the case could potentially be appealed to the United States Supreme Court.

For example:

If the appeal is based on a misinterpretation of the elements of a crime, the issue is with the state government. If the trial judge convicted someone of burglary, when the elements of burglary were not met, the state supreme court would be the final appeal. This is because the crime of "Burglary" is based on state law. The federal court system does nothave the ability to interpret state laws.

However, if the appeal is based on a violation of U.S law, the issue is with the U.S government. If the trial judge convicted someone of burglary, but the evidence was illegally obtained, the case could potentially be appealed to the United States Supreme Court. This is because the rules of evidence collection is based on federal law (The 4th and 14th Amendments to the U.S Constitution). The federal court system does have the ability to interpret Federal Laws (obviously).

So, in the first example, the case would start out in a

1. Trial Court (or a normal court)

Then, could potentially be appealed to the

2. State Appellate Court

And, afterward, may potentially be appealed (again) to the

3. State Supreme Court

Whereas, in the second example, the case would start out in a

1. Trial Court

Then, could potentially be appealed to the

2. State Appellate Court

Then, potentially the

3. State Supreme Court

And the final appeal could be toward the

4. United States Supreme Court

However, keep in mind that these appeals are not automatically accepted. The appellate courts are certainly capable of rejecting the appeal.