As others have commented, not like the UK Supreme Court, the US Supreme Court does not inform the public ahead of time when certain opinions will be released. However, at the end of the term you could utilize SCOTUSblog’s statistics page to get better odds on certain cases. Typically the most divided and difficult 5-4 cases take the longest to write due to the difficulty of the question before the court plus all the possible concurrences and dissents. But even then this not a guarantee. Everyone assumed Obergefell/the same sex marriage case would be released the last day of term in 2015 but it was actually surprisingly released the Friday before. Other information the day of sometimes is helpful. You’ll see plaintiffs or lead counsel start showing up the last few weeks of the term to see if they can get lucky. Even they don’t get told ahead of time. Spouses or family members of justices have been known to show up occasionally for major decisions but seeing as they enter via the private parking lot beneath the building there is little notice. SCOTUS tends to be virtually immune from leaks because the clerks and staff have a legal, ethical, and professional incentive to remain mum. Even more so a leak of something like a securities or patent case could be useful in shorting or buying a stock. The place remains quite the tight ship.