Beginning with schools officially operated by Church Educational Services, LDS Philanthropies shows that the Church currently operates LDS Elementary School in Suva, Fiji as well as the Fiji LDS Church College, which is really the equivalent of an American high school (most of the Church schools in the Commonwealth nations follow this convention of referring to secondary schools as "colleges").
The Church operates Moroni High School in Kiribati.
In Tonga, the Church operates Saineha Middle and High School on Vava'u; Liahona, Pakilau, and Havelu Middle Schools, and Liahona High School on Tongatapu, and middle schools on Eua and Ha'apai.
In Samoa the Church operates the LDS Church College in Pesega, LDS Church College and Primary School in Vaiola, and the Sauniatu Primary School.
The only other schools for which the Church advertises its support are the Juarez Stake Academy or La Academia Juárez and El Centro Escolar Benemérito de las Américas, both of which are the equivalent of high school programs.
While the Church-run schools in the South Pacific are perhaps the most famous schools in the recent history of the CES, the Church policy on education, ever since the call to gather to Utah was ended, has been one of building schools only where local educational opportunities were insufficient to meet the needs of the members. With the recent shut down of the Church College of New Zealand, as well as earlier shutdowns of Church primary and secondary schools in Mexico and elsewhere (I've have heard unsubstantiated reports of former Church schools in Peru, Chile, and Bolivia), one might be led to believe that it is only a matter of time before the Church pulls out of primary and secondary education entirely. However, it is telling that while some primary schools were shutdown in Mexico, the Church still operates two high schools in that nation, one of which is well over a hundred years old. Even as the local education systems continue to improve, the Church schools have become a part of the local community, and pulling out is never easy. (One need look no further than the reaction to the shutdown of the Church College of New Zealand.) While the Church may be hesitant to open new schools, they will also likely be slow to close them down.
While the Church has been slow to build more schools, occasionally one will hear stories about partnerships it has entered into with members who already owned schools. I've heard of at least two stories, one in India, and one in South Africa. Hopefully I will be able to pull more information in the future, but in both of these schools the founder was a convert to the Church who built the schools primarily to help orphans and low-income students. The Church agreed to provide some financial assistance, and with no pressure from the Church, the owners introduced gospel instruction into the curriculum. This would be a good example of a project where the Church influence is directly felt, but it nonetheless tries to keep a low profile.
The next category of schools would be the independent LDS schools, i.e. those like SVU which have an LDS mission, but have no formal ties with the Church.
The American Heritage School in American Fork, Utah mentions the "restored gospel of Jesus Christ" in its mission statement directly. This clearly qualifies it as an independent LDS school, but it has also lent its name to a series of independent "sister schools" known as American Heritage School of Las Vegas, Cache Valley, South Jordan, and Spanish Fork.
The Spanish Fork campus moved and changed its name to the Helaman Academy. It quotes extensively from LDS scriptures in its mission statement and references a commitment to "the gospel" (emphasis added), which would definitely qualify it as LDS in character.
The Las Vegas Campus and South Jordan Campuses fail to mention the Church in their mission statements, at best claiming to be "Christian", but both schools explicitly adopt LDS values in their code of conduct, so it's safe to say they are LDS in nature if not explicitly in name or purpose. The South Jordan campus appears to be the third iteration of a school, previously known as Mount Hyrum, which was more explicitly LDS, whereas the Las Vegas campus appears to have grown directly out of an LDS homeschool group.
The American Heritage School of Cache Valley was apparently a reincarnation of the Liberty Education Center, which was a reincarnation of the Kimber Academy, but it does not appear to be in operation anymore. The Kimber Academy on the other hand is operational from all appearances, and has affiliates in Lehi, UT; Midvale, UT; Moses Lake, WA; and Boise, ID. (I can only find evidence that the Lehi and Midvale affiliates are currently operational.) The Kimber schools don't appear to reference the Church directly, either in a mission statement or the code of conduct, as did the American Heritage Schools, but the Book of Mormon appears to be an integral part of the Kimber curriculum.
As was mentioned briefly before with regards to the Las Vegas campus of the American Heritage Schools, it appears that many of the LDS schools have grown out of or been given support by small LDS homeschool play groups, which seems an appropriate place to mention a growing phenomenon nationally that has been particularly evident among Evangelical Christians and more conservative Latter-day Saints. That phenomenon is a preference for homeschooling. Among the Evangelicals, the number of home schoolers has grown so significantly that they now sponsor the Home School Legal Defense Association (HSLDA) on a shared campus with Patrick Henry College (a college that was founded in 2000 largely to cater to homeschoolers) in Purcellville, VA. The Latter-day Saints are not quite as well organized, but several resources and organizations have grown up to support members who choose to homeschool. These include: the LDS Home Educators Association, the National LDS Homeschool Association, the Latter-day Saint National Home Educators, the School of Abraham, the Washington State Latter-day Saint Family Educators, LDS Homeschooling in California, the Latter-day Saint Eastern Home Educators (in which my sister actively participates), and the Maeser Academy (an online LDS high school).
Something fascinating, but not particularly surprising considering LDS culture, is the way the different LDS education initiatives interact with each other. For example, Latter-day Saint Eastern Home Educators have held their annual conference at Southern Virginia University for the past couple of years, and though separated from some of their peers in the west, the Latter-day Saint Eastern Home Educators discuss and debate the methods and resources coming out of Utah more than the amble resources which are so much closer in Purcellville.
One of those topics of discussion is the Thomas Jefferson Education. The entire LDS homeschool network seems to have been affected in one way or another by the Thomas Jefferson Education, so much so that many individual homeschoolers are defined by whether they support or oppose the method, albeit the larger organizations and networks tend to be neutral and thereby more inclusive. One even sees the influence of a Thomas Jefferson Education at institutions like the original American Heritage Academy, which is older than the method.
The Kimber Academies and many of the homeschool groups also appear to have close ties to another institution, George Wythe University, whose founder first articulated the Thomas Jefferson Education. This school was not mentioned in the entry on LDS higher education because it does not claim to be LDS, but clearly most of the students, the founders, and supporters are LDS. It will be interesting to see how George Wythe University and Thomas Jefferson Education continue to shape the future of LDS education. This movement has certainly had the most success at establishing independent LDS educational institutions in recent history.
When one moves away from this particular philosophy of education it is difficult to find any other successful independent LDS schools other than those previously mentioned in India and South Africa where the Church has had some direct involvement. I am aware of an LDS boarding school founded in Japan, but it barely lasted four years before going under. The Deseret Academy in Utah is another example of an LDS school that just couldn't survive.
As I become aware of other schools, I'll try to list them, but for now this is what I could find.