One of the most important splits in Methodism came as a result of slavery. This occurred in 1844 and accompanied similar schisms in the Presbyterian and Baptist churches. It occurred when Methodist leaders resolved to censure one of its members for slave ownership and actually led to further splits, as antislavery members from southern states (a small minority by the 1840s) split away from the main southern body. This exemplified the depth of the moral crisis of the period and the ways that evangelical faith was used to prop up, as well as critique, slavery.
In the 1950s and 60s, there was significant disagreement among Methodists over how to respond to the movement for civil rights in the South. Some Southern members openly supported segregation. Others ostensibly sympathized with the aims of the movement, but not the tactics employed by civil rights protesters. These were among the "white moderates" criticized by Martin Luther King in his "Letter From a Birmingham Jail." Some northern Methodists, on the other hand, openly supported the movement, including its tactics. They openly embraced the nonviolent approach advocated by King, marched in such events as the March on Washington, and responded to King's "call for clergy" at Selma.
In the 2010s, profound disagreements exist among Methodists over LGBTQ rights. The question of LGBTQ clergy has been especially controversial, with a recent vote by United Methodist leaders forbidding both same-sex marriage and the ability of LGBTQ Methodists to serve as clergy in the church. Some conservative Methodists have even advocated denying church membership to LGBTQ men and women. The issue remains one of the most profoundly divisive in international Methodism, with the divide mostly occurring between rural and urban members, and between American contingents and those from overseas. Southern Methodists, however, tend to be more conservative on these issues than their northern counterparts.
In short, Methodists are, like all groups, subject to the same social divisions as society at large. Despite their theological agreements, they differ on how to apply their theology to secular issues.