The Supreme Court of the United States heads the most publicity shy branches of government. The politicians in Congress need publicity for reelection, and many seem to crave it. The President has no choice but to live in a fishbowl. But the Supreme Court Justices are appointed and have no need for reelection or popularity. Some probably shy away from publicity because they are naturally scholarly types, and others do so because judicial ethics and customs put a tight rein on what judges say in public. To date, there is no videotaping of arguments, and the Justices have not looked kindly on lower courts’ admission of cameras into the courthouses.
Even with these limitations, though, the Supreme Court building is open to the public, and welcomes anyone to attend oral arguments on a first come-first serve basis. The building itself is magnificent, and the courtroom especially is awe-inspiring (and not very large). Regardless of whether one is inclined to visit, the newly redesigned Supreme Court website is terrific.
Information to guide visitors provides thorough information about the argument calendar, what cannot be brought into the building, and how to get to the court. Without leaving your desk, researchers, or just interested citizens, can find lots of information including opinions, transcripts of arguments, and sources for briefs. (The Supreme Court link for recent briefs is not working as of this writing.)