This is admittedly an educated guess and may be applicable only in the US: But I believe the answer would be Municipal Courts or County-level court systems probably hear the most cases due simply to the sheer number of misdemeanor and traffic cases that are heard each day.
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It can depend on what kind and type of cases you are referring to. Probably the most overwhelmingly busy of all of the courts is Traffic Court. Other than that, county level and state circuit courts are probably the most busy level of the court system.
federal court
The Supreme Court is brought a large number of cases every year to review, but they do not have to hear all of them. They choose from cases that have already gone through state or federal courts where one of the parties was unhappy with the previous decision.
The Supreme Court of the United States was created in 1789. Most of the cases the court hears come from lower courts. Each year, the Supreme Court receives 7,000 or more requests to hear cases from lower courts.
evaluate the decisions of lower federal court
US Court of Federal Claims would hear those cases.