Some courts can pick and choose the cases they are going to hear, at least to some extent. Others cannot. Of the on's that can, the first task of advocacy as to such a court is to convince it to hear the case in the first place. A common rookie mistake for appellate lawyers is to try to sell the court on the basis of the enormity of the injustice below. One practically always does much better to show that the courts below are following different, inconsistent rules as to those kinds of cases, or, if that is not possible, to show that the court below did not follow the precedent of the court one is trying to get into.
What kinds of courts can pick and choose their cases? Well, except for certain types of cases mandated by the U.S. Constitution, the Supreme Court of the United States chooses which cases to hear by a vote of their members. A case that four or more of the justices want to hear gets heard. A case that less than four of them want to hear does not get heard. The Supreme Court of Texas reviews cases that four of the justices want to look at. The Texas Court of Criminal Appeals has to hear all appeals of all death cases, but otherwise, four votes or more get a case heard. Federal trial courts don't have to hear certain types of cases-- family law, for example, but the federal courts of appeals pretty much have to hear every case the federal trial courts and the agencies send up to them.
Courts that get to choose what they hear are generally more interested in standardizing the precedent used by the courts below them, than they are in correcting gross miscarriages of justice. Such courts often have judges or court below them that they are inclined to correct. SCOTUS is inclined to correct San Francisco's Ninth Circuit. SCOTX, the Thirteenth Supreme Judicial District of Corpus Christi and Edinburg. Texas's Court of Criminal Appeals, judges who hold pretrial hearings on the constitutionality of the death penalty.
Such courts are unlikely to take a case the first time an issue arises; they want to have the lower courts wrestle with it. If the lower courts agree on how to handle a matter, they've just saved the higher court pointless work. A selective higher court would generally prefer to take a case that they have the votes to overrule; that way, they don't just parrot the lower court. Realize that a higher court judge who disagrees with a lower court decision may not have the votes on his or her court to overturn it. Such a judge may not vote to hear such cases, preferring bad lower court decisions to bad higher court precedent.