Why the interest in locally Euclidean spaces? A lot of mathematics as far as I know is interested in the study of Euclidean and locally Euclidean spaces (manifolds).


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*What is the special feature of Euclidean spaces that makes them interesting?

*Is there a field that studies spaces that are neither globally nor locally Euclidean spaces?

*If such a field exists, are there any practical uses for it (as in physically existing models)?

 A: I'm interested in differential topology, so that baises my answer.  $\mathbb{R}$, and Euclidean spaces generally, are interesting because you can do calculus on them.  Manifolds are interesting because, not only as OrbiculaR says, you can do standard analysis in a chart, but you can sensibly extend the concepts of analysis to the entire manifold.
You can also do analysis in the p-adics, so you might wonder why aren't studying p-adic manifolds.  (I don't actually know.)
A: One of the main reasons for interest is that manifolds are homogeneous.  So from a geometric point of view, we study manifolds for much the same reason we study groups.  
Put in less abstract ways, there is the "classical problem" of whether or not the earth is flat.  The "flat earth problem" deals with the possibility that on small scales something may look linear, but macroscopically it need not be.  Manifolds are the abstract manifestation of the flat earth problem. 
People do study large families of objects that are locally modelled on spaces that aren't Euclidean spaces.  There's infinite-dimensional manifolds, fractal manifolds, orbifolds (locally modelled on euclidean space mod a finite group action), etc... Fibre bundles are maps that are locally projection maps.  This idea resurfaces in mathematics in many different forms and many different ways. 
A: In my opinion (because it really is an opinion), the main reason to care about euclidean spaces is that they come equipped with a lot of particularly nice structures.  In particular, they are finite-dimensional Hilbert spaces.  This means that they are:


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*Finite-dimensional vector spaces (so we can talk about addition and scaling)

*Complete metric spaces (so we can talk about distances and limits, and limits behave as we would like them to)

*Inner product spaces (so we can talk about the notion of "angle," and thereby do all sorts of geometric things.)


And really, there's just so much geometric intuition that comes along with these ideas -- not to mention an entire calculus apparatus.  After all, there are plenty of topological spaces where the above notions are not defined or otherwise fail to be true.
The reason (at least to me) to study locally euclidean spaces is that we want to study spaces that are more general than euclidean spaces, yet still retain many of their nice features.  In particular, we want a place where calculus makes sense.
Areas of math which study more abstract spaces include topology and algebraic geometry.  Admittedly, I'm not very well-versed in either just yet, but I'm sure practical uses (and physical models) have been found in both.
A: Well, my answer to 1. is that you can convienently do standard analysis in a chart. Did you ever try working on more complicated (singular) objects? It's a pain in the neck! (2. and 3. have already been answered)
A: In topology people study intensely some naturally occurring abut quite abstract and not "locally Euclidean" spaces. Examples are path spaces(space of paths in a given topological space or manifold) and some natural $H$-spaces in the homotopy category.
Anyway there a whole lot of new kind of "geometry" in the field of algebraic geometry, which is not at all Euclidean.
But your point is good, that the Euclidean space is quite specially useful in physics. With this in mind, some people are actually doing adelic physics these days.
A: I think the interest is historical. Because mathematics and physics grew up together, physical continuity† could easily be argued to be of mathematical interest & relevance.
† Remember how surprised physicists were when discontinuous "jumps" were found! (The quantum revolution.) iirc this was first proved with black-body radiation experiments.

I can't find the book I'm trying to quote from right now (and will try to edit this later if I remember), but chapter 1 or 0:

Manifolds historically were always understood to be some subset of ℝⁿ. It was only in the early 20th century [let's say due to Noether or Klein] that people decided we could do away coördinate charts altogether, and the notion of an abstract manifold (now the familiar one) was introduced.

You can find this also in Spivak DG1, where he refers to eg pushing forward a tangent bundle as the "modern, clean" way of doing things --- which is less intuitive, but also in a sense preferable, to the older "subset of ℝⁿ" way of seeing things.

You asked if people do study non-smooth spaces. I am not expert enough in any of these to really give an answer, but try googling on:


*

*pro objects

*p-adics

*paracompactness

*non-Hausdorff


Usually non-Hausdorff would be considered pathological – so I think it would take a beautiful result or another kind of compelling vision to get people interested specifically in some non-Hausdorff object. But one man’s pathology is another’s "very interesting": Dedekind, Cantor, Sierpiński, and Gödel are just a few who had a great time and earned plaudits studying spaces that I consider pathological. AMS obits can be a decent source of information about how mathematicians come to find various spaces / objects interesting. (I have one in mind specifically about Polish spaces, but whose obit it was slips my mind at the moment.)
By "pathological" I mean that I think a lot of mathematicians would be thinking either "Why? That never happens" or "Does anything interesting happen if you start there?"

Finally, so-called "arithmetic manifolds" may be an example in the intersection of mathematics that is taken seriously ∩ not locally Euclidean. The Weil conjectures; some ideas of Peter Sarnak; Tao-Green theory (arithmetic progressions) ‒ require alternate cohomology theories like étale or crystalline – and within the construction of those there might be an assumption of non-smoothness (eg, the primes are not smoothly distributed). Using less technology, you might be able to google on L-functions (lmfdb.org), weights, or elliptic curves (I believe Silverman has an NSA-crypto style introduction assuming less vocabulary). If you search up "Verdier duality" you will see again people trying to get something like Poincaré duality, but for non-manifolds.
Hope that answers your question! (And assuming others who share it will find this page, in case you aren’t checking the site anymore.)
