Your question is somewhat misleading, so I am not sure that my answer is appropriate; however I will attempt to explain.
Most criminal cases are resolved by plea bargains. Contrary to that which appears on television programs, evidence of guilt in most cases is overwhelming; proceeding to a jury trial would be both wasteful and dangerous. Those who insist on jury trials in the face of obvious guilt normally receive more harsh sentences as this is considered a refusal to accept responsibility for one's actions and thereby lack of remorse. Only those cases in which the penalty is especially harsh or there is some question of a guilty verdict will a defendant normally choose to "roll the dice" with a jury; most choose to cut their loses and plead.
The tricky part of your choices is that even defendants who plead guilty must do so before a judge. No sentence can be imposed until a defendant has pleaded to the charge in open court. Since all guilty pleas are heard by a judge sitting in court, District Court is the correct answer. I must say, however, that your question is poorly worded. In New York, the "Supreme Court" is the trial court which hears guilty pleas. Municipal and county courts exist everywhere, and district courts normally imply Federal District Courts, although some states divide their judicial system into judicial districts.