An appeal to the Supreme Court or to a court of appeal from a judgment of conviction stays the execution of the judgment in all cases where a sentence of death has been imposed, but does not stay the execution of the judgment or order granting probation in any other case unless the trial or appellate court shall so order. The granting or refusal of such an order shall rest in the discretion of the court, except that a court shall not stay any duty to register as a sex offender pursuant to Section 290. If the order is made, the clerk of the court shall issue a certificate stating that the order has been made.