Among the many, many reasons why it is crucial for our next President to be Hillary Clinton and not Donald Trump, the chance to finally put together a progressive majority on the Supreme Court (for the first time since arguably the Nixon administration) is among the most essential. The next President will appoint the replacements for the late Justice Scalia, and almost certainly Justices Ginsburg and Kennedy as well, offering the opportunity for a 6-3 Democratic majority (or, conversely, a 6-3 Republican one). The report that Justice Thomas is also mulling retirement would be icing on the cake, and while at this point its pretty thinly sourced, like Justice Souter Justice Thomas never struck me as the sort of person in love with the job, and so I can imagine him retiring earlier than some of his fellows.
But what would be the most likely alterations we'd see from a progressive Supreme Court? Political Scientists have long believed two things about the courts vis-a-vis political society writ large: (1) They tend to follow the prevailing political trends, at least those with traction amongst the elites of their ideological cohort, and (2) They're limited in what they can accomplish without the backing of the political branches.
Many items high on the progressive wish list would have limited purchase in absence of legislative control in Congress. A decision overturning or circumscribing Citizens United or Heller would permit Congress to pass more robust campaign finance or gun reforms (respectively), but of course it wouldn't force Congress to do so. For the most part, courts can clear the brush for political actors, and can play defense against hostile legislative encroachments, but they are quite limited in their ability to move the ball forward on their own.
But one area where I think we may witness significant changes is in criminal justice issues -- searches, seizures, sentencing, and qualified immunity. These are areas where we are seeing a significant shift in the political terrain, at least amongst the left, and likewise are more directly under the judicial umbrella than many other areas of social reform. A Court which is less keen on believing that police brutality suits are simply frivolous harassment by social bottom-feeders is likewise less likely to give qualified immunity an expansive reading. A Court which stops seeing unlawful searches as the overzealous effort to preserve social order, and starts seeing it as a threat to social order, will do more to curb those searches and check unlawful behavior by police departments.
Justice Sotomayor's broadside dissent in Utah v. Strief, which clearly has been influenced by changes in the zeitgeist brought about by movements like Black Lives Matter, represents the tip of that particular iceberg. A Supreme Court majority which has that outlook towards the criminal justice system could bring about reforms to a degree unrivaled since the Warren Court era. Law and order, after all, includes the obligation on the state and its agents to stay within the law and follow judicial orders. This election will do much to determine the degree to which everyone -- officer and citizen alike -- are made to live under the rules set by our constitutional system.