WASHINGTON  The new Supreme Court term, which begins Monday, includes cases on some of the most contested issues of the day, including protests at military funerals, illegal immigration, support for religious schools, violent video games, DNA evidence and prosecutorial misconduct.

The term’s arguments and decisions will be scrutinized for insights into the thinking of the court’s newest member, Justice Elena Kagan, and for hints about how the court will rule when even more highly charged cases reach it, probably in a year or two, on federal health care legislation, same-sex marriage, the treatment of gay members of the armed services and the recent Arizona law giving the police there greater authority to check the immigration status of people they stop.

The marquee case on the docket so far is a suit brought by the father of a fallen Marine against a small Kansas church whose members protested at his son’s funeral. The case, to be argued Wednesday, is freighted with rage on both sides.

“Since when did any of our military die so that a group of people could target their families and harass them?” asked the Marine’s father, Albert Snyder, who won an $11 million jury verdict against the Westboro Baptist Church of Topeka, Kan., saying the church had caused him emotional distress.