Elena Kagan's recent nomination to SCOTUS has brought attention to Jews and the Supreme Court - if confirmed, she would be not only the third woman (in history) but also the third Jew (sitting now) on the American Supreme Court, making that institution one third Jewish!
(She was also, through her own struggles, the first girl to have a bat mitzvah at New York's (Orthodox) Lincoln Square synagogue.)
What else do Jews have to do with the courts?
The American Jewish Committee has been filing 'amicus curiae' briefs in the Supreme Court since 1923. AJC in the Courts:2008 reports on their briefs filed on cases relating to separation of church and state, civil rights and civil liberties (including gun control, reproductive rights, and school integration), religious liberty, and Holocaust restitution.
I don't think anybody will be shocked at the generally liberal positions taken by the AJC, though they are not entirely uncontroversial. Yossi Prager's recent article, Day School Sustainability: Ours to Achieve, disagrees with their position on government funding for religious education, for example.
For more details, the AJC's litigation reports from the last decade or so are available on BJPA. Each report summarizes the case and then the position taken in the AJC's brief.