What issues does the North American* LDS Church face in the 21st century? Surely a host of them. But here are two that come to mind.
What about women in the Church? The press that Ordain Women received made that movement's issues the ones we think most about when we think about women in the Church. We less often think of the other, often more pressing, issues that LDS women face and how we are or are not dealing with them. If we ignore the question of women's ordination, it is important to recognize that, sometimes in spite of the body of the Church, the leadership has done well. They have brought women into more prominence in General Conference and have urged for more participation by women in the leadership of wards as well as the planning of Sacrament meeting. It is clear that they recognize the problem and are working on it.
But the question remains of what else needs to be done. For example, we say that the ward council should take an active role in planning Sacrament meetings, but how do we make that happen in practical terms? In spite of the general leadership's urgings, many wards have yet to take seriously the requirement to meet in council and to allow all its members to speak openly and equally. Too often councils, where women could in principle have a prominent role, are merely a formality and too often they are no more than an "amen" to what the bishop proposes. That, too, needs to be addressed. Further, we don't yet do much for never-married, divorced, and widowed women. If families are forever, where do they fit? The question may be answered in eternal terms, but in day-to-day terms, we have hardly begun to think about it. What else remains for women?
What is the Church's culture? Since the early 20th century, after the exodus of the largest part of the Church from Britain to the U.S., church culture has been primarily that brought by the missionaries. In other words, it has been primarily North American and particularly that of Utah. Many LDS outside North America have begun to develop their own variations on that culture, but those variations always exist in tension with the culture of the missionaries and expatriate members, who are primarily North American.
Because the LDS Church is more focused on practice than on theology, some common culture is essential, even if it has variations. Mormons share a common history, even if it is an adopted history for most of us, and we share a common culture. We also share a common set of core beliefs, but our history and culture give us our identity more than our common beliefs.
Presently North American/Utah culture dominates in the Church even when it need not because we have rarely thought about the question. As a result, sometimes we alienate people unnecessarily and unconsciously. It is natural to North American Saints to think that our way is the way. Thinking about what is essential and what is not will almost certainly require changes in the culture and attitude of the North American Latter-day Saints and perhaps others as well. This is tricky because so much of culture is unconscious, but tricky or not, unless we think more explicitly about what is necessary to our culture and what is not, North American culture will continue to dominate the world-wide Church. How does LDS culture in North America need to change in response to that problem?
Of course there are other things we need to face, but to my mind these are among the most prominent. Whatever problems we have, though, unlike some I'm optimistic for the long run. Having made it through the 19th and 20th centuries, I'm confident we'll find our way through this one.
* I here use "North America" as a shorthand for "English-speaking U.S. and Canada," which is part of the problem: like most in the U.S., we seldom recognize the non-Anglophonic parts of North America, like Mexico and the Caribbean, much less the non-Anglophones among us.