The Supreme Court’s ruling Monday on the religious rights of private for-profit businesses ends two cases that have garnered loads of national attention because they involved President Obama’s Affordable Care Act, one of the symbols in 2014 of partisan divide.

Hot-button as it was, however, the issue often short-handed as “Hobby Lobby” — after the evangelical-owned crafts chain that was one of the parties suing the White House — is just one sign of many that the relationship between church and a much more diverse American state is being renegotiated.

To some it feels like the place of religion is being demeaned or threatened, but church-state experts say the truth is more subtle, that U.S. society is simply becoming more complicated, and that major cases today sometimes boost religion and sometimes limit it.

“If there was ever a time when it was possible to talk about a separate sphere, that time has long since passed. So now we have to ask more nuanced questions,” said Robert Tuttle, a church-state expert at George Washington University Law School and an attorney for the Evangelical Lutheran Church in America. “We don’t have a wall of separation between church and state. Religious organizations already get funding and are subject to lots of regulations already. Today we have to learn to make finely grained judgments.”

Here are four areas of church-state conflict to watch:

●Pensions: Some of the country’s largest Catholic hospital chains are facing lawsuits by employees who say the chains have illegally exploited tax loopholes for churches that allow affiliated institutions to avoid insuring or fully funding pensions, among other perks. The cases involve tens of thousands of employees and may potentially affect the health of Catholic hospitals in particular.

Non-Catholic institutions are also coming under some criticism for seeking the exemptions, which were created in the 1970s as part of the passage of the Employee Retirement Income Security Act (ERISA). The exemptions were meant to keep the government from intruding into the financial affairs of houses of worship and other clearly religious groups. But over the years more and more institutions have been claiming the exemptions, such as schools or secular hospitals that are affiliated with churches. Tax law says an organization is associated with a church — and qualifies for an exemption — “if it shares common religious bonds and convictions.”

In some cases these exemptions have meant billions of dollars in plans that are exempt from ERISA protections, said Karen Handorf with the firm Cohen Milstein, which is involved in many of the lawsuits underway.

“We call these ‘Hallelujah Plans’ — they found the Lord,” Handorf said. “Some of these would be Fortune 500 companies if they weren’t nonprofits.”

The bulk of the suits have been filed in 2013 and are moving forward.

The issue came to public light now, Tuttle says, because of the economic downturn of the late 2000s. Church plans were unable to meet their pension requirements and didn’t have ERISA protections. Pension advocates became concerned about the health of the plans.

●Parsonages: There are just two categories of people who get a special housing exemptions on their income taxes: military members and pastors.

But now the “parsonage exemption” is being challenged in federal appeals court after a Wisconsin judge in the fall said the benefit is unconstitutional.

The exemption “provides a benefit to religious persons and no one else, even though doing so is not necessary to alleviate a special burden on religious exercise,” U.S. District Court Judge Barbara Crabb wrote in November.

This would strongly impact tens of thousands of clergy who are allowed to set aside from their federal income taxes money related to housing. Tuttle said some clergy get even 50 percent of their salary as a housing allowance.

Many of the federal tax benefits religious organizations get are because they are 501(c)(3)s — or nonprofits, and are the same benefits other charitable or educational institutions get, such as museums or zoos. But housing exemptions are just for members of the military and clergy, Tuttle said.

The case is being brought by the Wisconsin-based Freedom from Religion Foundation, a nonprofit.

The U.S. government is defending the exemption, saying it is unconstitutional meddling in church affairs to determine how much housing compensation clergy might get, Tuttle said. The case will continue to push the broader question all of these topics face: What is the rationale for separate spheres for the religious and non-religious?

●Public schools and churches: The Supreme Court earlier this month declined to hear the case of an Illinois school district that was challenged for holding its graduation in a church, complete with a massive cross. But the issue of how public schools and religious institutions mix is very much alive.

The center of the dispute in recent years has been New York, where the city’s Department of Education bans church services from being held (off-hours) in public schools. A federal panel in April said the policy is constitutional. Many churches around the country use public-school buildings on weekends or after school.

The church at issue — Bronx Household of Faith — argued that the city was violating the First Amendment’s guarantee of the free exercise. The backdrop is the city’s sky-high real estate costs, and the church said it needed an affordable space. But the federal panel of judges wasn’t convinced by the district, or lower, court’s thinking.

“In the District Court’s view, because Bronx Household and its congregants have a constitutional right to worship as they choose without interference from government and cannot afford to pay for a large enough site to accommodate the entire congregation, the Free Exercise Clause obligates the [department] to provide them with a subsidized facility in which to exercise the right,” two members of the three-judge panel of the U.S. Court of Appeals for the 2nd Circuit wrote. “The Free Exercise Clause, however, has never been understood to require government to finance a subject’s exercise of religion.”

New Mayor Bill de Blasio said he would change the policy and allow churches to use the school buildings — but it wasn’t clear when that would begin.

●Chaplains: Earlier this month we covered the story of Jason Heap, a Protestant-youth-minister-turned-Humanist, whose application to be a Navy chaplain was rejected. The Navy didn’t detail to Heap — nor to The Post — why he was rejected, but no branches of the U.S. military currently have explicitly non-theistic chaplains.

But the dispute over chaplains, particularly in the military, takes many forms.

One of the more closely watched cases in recent years involves evangelical Protestant Navy chaplains who say they are discriminated against in favor of chaplains who are Catholic or who come from liturgical Protestant faiths such as Lutherans or Episcopalians.

The plaintiffs argued in court that they are not treated equally when it comes to things like promotions, but other lawsuits against the Navy have raised a broader complaint: That the evangelical chaplains were being discriminated against for not being more “pluralistic” — for example, by insisting on prayer in the name of Jesus when invited to speak at official events.

For many evangelicals, personal witness to Jesus is basic to their understanding of faith and practice of prayer.

The evangelical cases are related to the humanist ones, Tuttle said, in that they both raise difficult questions about how the government allocates positions in the chaplaincy.

“Can it decide that one group is not sufficiently religious, because it has no belief in an afterlife or supreme being, or that another group is ‘too religious’ in that it can’t effectively minister to other faiths?”