The cases that typically draw the most attention touch on novel issues of law with sparse precedent that allow the justices more room for disagreement or those with significant political ramifications and broad interest in the public square. Think, for example, of past legal and political battles involving abortion rights or affirmative action. Below are a handful of cases this term that are likely to grab the most headlines.

Florida v. Department of Health and Human Services. No discussion of the upcoming term can begin without talking about a case that isn’t formally on the docket yet. Last week, the Obama administration asked the justices to review an appeals court decision that struck down as unconstitutional the centerpiece of the Obama health-care plan: the individual mandate, which penalizes individuals who do not purchase health-care coverage. It is almost a given that the justices will agree to hear this case or one of several other challenges to the health-care law. What is not clear is whether the justices will tackle the constitutional issue and decide whether the Constitution’s commerce clause gives the federal government the authority to mandate coverage or whether they will decide the case on narrower, technical terms, such as whether certain parties have the legal right to challenge the law. Although the individual mandate will not kick in until 2014, insurance companies, health-care providers and individuals are already preparing for the event. A high court ruling now would bring clarity and potentially save all those who are affected a lot of time and money.

Douglas v. Independent Living Center of Southern California. Scheduled for argument on the first day of the term, this case centers on the efforts of some Medicaid providers in California to challenge the state’s decision to cut the rates they are paid. But the justices won’t be deciding whether or not California was right to try to gets its fiscal house in order by reducing Medicaid costs. Instead, they will determine only whether the providers who brought the challenge had the legal right to do so. This is a much narrower and far less sexy legal issue but one that could have broad ramifications for all manner of challenges going forward.

Maples v. Thomas. Corey Maples has never claimed he was innocent of the murders that landed him on Alabama’s death row. But he has claimed that he should not have been hit with the ultimate penalty. But Maples’s chances for challenging his sentence were dashed because of a clerical error. A critical notice that was sent to his New York lawyers was returned to the Alabama courthouse because the lawyers had failed to notify the court that they had left their firm. An Alabama lawyer assigned to the case did not take action, assuming that the New York team would continue leading the defense. The Alabama clerk’s office also did nothing. The justices must decide whether Maples should get another shot at appealing his sentence or whether he will face execution because of his lawyer’s mistakes.

Hosanna-Tabor Evangelical Lutheran Church and School v. EEOC. Religious organizations are allowed to discriminate in making hiring decisions — and rightly so. After all, it would be absurd to force a synagogue to hire a Christian minister as a rabbi. But does that same protection against discrimination lawsuits apply when it involves employees who are not religious leaders such as rabbis, priests and pastors? That is the question before the Supreme Court, which has been asked to decide whether a teacher at a small Lutheran school may proceed with her suit alleging that she was dismissed because of a disability.

Florence v. Board of Chosen Freeholders of the County of Burlington, N.J. Albert Florence was pulled over in 2005 on a routine traffic stop and arrested when the police officer became aware of an outstanding warrant for Florence for failing to pay a fine. Although Florence, an African American finance director at a car dealership, had a receipt proving he had paid the fine, the municipality had failed to remove the bench warrant from its system. He was jailed and twice subjected to invasive strip searches over the following week. The court must decide whether “the Fourth Amendment permits a jail to conduct a suspicionless strip search of every individual arrested for any minor offense no matter what the circumstances.” The Obama administration has sided with New Jersey, arguing that law enforcement officials must have maximum flexibility to prevent drugs and weapons from being smuggled into corrections facilities.

United States v. Jones. Antoine Jones was a Washington, D.C., nightclub owner and, police suspected, a drug dealer. So as part of their investigation, officers attached a Global Positioning Satellite tracking device on a Jeep driven by Jones. Officers had obtained a court order to do so but did not place the device on the Jeep until after the order had expired. Police used GPS to follow Jones’s every move for roughly one month. Jones was convicted by a federal jury but his conviction was reversed by a federal appeals court that determined that the Fourth Amendment prohibition against unreasonable searches and seizure required a valid court order for GPS use. The Obama administration has asked the justices to throw out this decision. It argues that law enforcement officers should have the same leeway to use GPS as they do now to tail suspects on public streets. This latter technique does not require prior court approval and neither should the use of GPS, the administration contends.