The vast majority of non-criminal cases in the United States are handled in state courts, rather than federal courts. For example, in Colorado, in 2002, which is typical, roughly 97% of all civil cases were filed in state courts and 89% of the cases filed in federal court were bankruptcies. Just 0.3% of the non-bankruptcy civil cases in the state were filed in federal court. In Colorado, in 2002, there were 79 civil trials in federal court (41 jury and 38 non-jury), and 5950 civil trials in state court (300 jury and 5650 non-jury). Essentially all probate and divorce cases are also brought in state court, even if the parties involved live in different states. . . .
About 91% of people in prison at any given time in the United States were convicted in state court, rather than federal court, including 99% of defendants sentenced to death. Federal courts disproportionately handle white collar crimes, immigration related crimes and drug offenses (these crimes make up about 70% of the federal docket, but just 19% of the state court criminal docket). A large share of the violent crimes that are prosecuted in federal court arise on Indian Reservations or federal property, where state courts lack jurisdiction and tribal court jurisdiction is usually limited to less serious offenses.
The vast majority of cases above the small claims court level, criminal and civil alike, never go to trial. Many cases are never appealed (although there are comparable numbers of appeals and trials because many cases in which judgments are rendered prior to trial on motion practice or due to a plea bargain where a sentence is disputed are appealed). Most cases appealed are affirmed.
It is also worth considering that relief from judgment of state courts is extremely rare, beyond the state court system. U.S. Supreme Court review and habeas corpus review, combined, for example, overturn only about 250 state court judgments per year for the entire United States. Pardons and commutations of any kind are likewise very rare, as are private bills that have the effect of overriding judicial decisions. There is considerably more state supreme court review of appeals of right (and in some states, appeal to the state supreme court is an appeal of right), but few cases that reach even that level in the judicial system.
Also, in the federal system, it is worth noting that only about 200 cases per year in the entire nation receive review on the merits beyond an appeal of right via en banc review at the U.S. Court of Appeals level, or in the U.S. Supreme Court, and that a significant share of those cases do not result in a reversal of a U.S. Court of Appeals three judge panel decision.