When can the US Supreme Court review a state supreme court decision?

A case can be appealed from a state supreme court to the US Supreme Court when there is a preserved federal question involved. A "preserved federal question" means the case raises issues about the US Constitution, federal law, or a US treaty at each step of the judicial process -- from trial through appeals.

Federal questions about the constitutionality of a state law may be raised and determined in state courts. If the state action goes to the state supreme court, but one party alleges that the state supreme court decision is wrong because it mistakenly interprets the US Constitution or federal statute, the US Supreme Court may consider an appeal from the state supreme court decision on a petition for writ of certiorari.

If the state supreme court decision is based entirely on the state constitution or state statute with no issue of a federal nature, then the US Supreme Court has no jurisdiction and may not hear the case.