Short Disclaimer:
I don't really have space to explain all of the elements of the plot in The Silmarillion, The Lord of the Rings, The Hobbit, and Tolkien's other books that take place in this universe, so I'm not going to (also, if you haven't at least read some of The Lord of the Rings or The Hobbit then you should be doing that instead of reading my blog post anyway).
Within the universe of J.R.R. Tolkien's trilogy
The Lord of the Rings, there are many civilizations that resemble our own in that they are neither utopias nor dystopias (interestingly, these tend to be the human civilizations). However, there are also several pockets of life that could be described as utopias. The most obvious forms of utopia in Tolkien's universe are those inhabited by the elves, who seem to live in continual happiness when they are not at war. In particular, in the elves' second home, known as Valinor the gods of Tolkien's universe (the Valar) live among the elves and they are all happy. The Shire is another Utopia that exists within this universe, and it is inhabited by Hobbits. There are not many dystopias in Tolkein's universe, and the few that do exist were created by Melkor (aka Morgoth who is basically Tolkein's version of the devil, and aside from stupid things that humans do, essentially the only reason anything goes wrong in Arda) or his servants.
There are two main kingdoms of Elves in Middle Earth during the events of
The Lord of the Rings, Lothlorien and Rivendell, though there were others before them. Both of them have several characteristics that I believe make them utopias. Aside from the fact that no one ever dies in either of these kingdoms (the elves are immortal unless they are murdered, and no enemy ever enters either kingdom), the two kingdoms exist in balance with themselves. Neither kingdom has issues with corruption, poverty, or disease, and both kingdoms are known for their prosperity and how they live harmony with the natural world. Most importantly however, all of the elves that live there are happy (when they are not at war with Sauron, which is most of their existence).
Similarly, in Valinor, the elves live in harmony with the gods of Arda (except Eru Illuvitar, this universe's creator who does not live on Arda). Again, elves are immortal unless murdered and the Valar are also immortal so no one dies in Valinor (which leads us to the nickname of the continent that Valinor is on: the undying lands). There are a bunch of different places in Valinor, which correspond to the realms of the different gods, but the important thing about them is that they're all paradises (except Mandos' area but that's essentially an afterlife. It's a little more complicated than that, especially for elves, but you get the point). Unsurprisingly, everyone in Valinor is also happy, and that's where the majority of the elves live.
The Shire is another place in Arda that is utopian. It is less of a utopia than the homes of the elves as hobbits are mortal (though they live much longer than humans), and they have petty issues, but it's still essentially a utopia. For one thing, the issues that they do have are extremely minor (almost always neighborly disagreements about noise levels, etc.). The prosperity of their society is well represented in the number of meals that all hobbits eat in a day--six. Hobbits are essentially ungoverned (there is a system of government in place but it really doesn't do anything and isn't required to), which means no corruption or systematic oppression. Poverty also doesn't really seem to be a thing that they have issues with even though there are different classes. Most importantly, again, everyone there is happy and content (which is why everyone thinks its so weird when Bilbo helps the dwarves in the events of
The Hobbit).
The kingdoms of humans (and there are a lot of them, though most of them collapse one way or another), are pretty normal societies. Human societies (like Gondor and Lake Town) have issues with corruption in the ruling class (like the Master and the Steward of Gondor). Their governments are much more oppressive of their people, and spend more of their time trying to maintain their power. Gondor goes through a number of civil wars due disputes over claims to the throne, and eventually the reign of Gondor is given to stewards because a proper heir cannot be discovered. Essentially, the kingdoms of humans function a lot like real human civilizations (not very utopian if you ask me, but the governments are not oppressive enough to be dystopian).
Dystopias also appear in Tolkien's universe, though there are not many. The kingdoms of orcs (also inhabited by trolls, etc.) seem dystopian, though we don't learn much about them. They live in places like Gundabad or Mordor, which are particularly desolate areas of Middle Earth. Orcs are elves who were corrupted by Morgoth, which basically means their only goal is to ruin everything for everyone else, including each other unless one of their leaders is present. Any servant of Melkor is essentially his slave, and is usually also overseen by more powerful servants. Places where orcs live have a lot of violence, slavery, and oppression for each other, and even worse for those who don't serve Melkor. They are also by far the most warlike civilization; the servants of Melkor cause almost every single war in Arda.
Although the main focus of all of the stories that Tolkien wrote about his universe is on the interactions between these civilizations, the people who inhabit them, etc., these civilizations could also be used to analyze our own in a similar way to those of utopian/dystopian novels. Though not all of it is realistic, I think it's instructive that the civilizations with less class divisions are more utopian, and so are the civilizations that are more peaceful. Humans and orcs are the two most warlike and most hierarchical groups, and they are also the most unhappy. Comparing that to our society is definitely disturbing, what do you guys think?