Are losing litigants always allowed to appeal their cases?

Are losing litigants always allowed to appeal their cases?
1) Yes, individuals that lose at trial court have a right to appeal to an appellate court. To appeal from an appellate court; however, requires a party to ask the Supreme Court to hear a case and the court decides which cases it accepts.
2) No, individuals do not get to choose to appeal, only judges decide which cases are worthy of appeal
3) Yes, individuals have a right to be heard at both an appellate court and at the Supreme Court,