How do we know that spacetime is uncountable/continuous/etc.? Every major theory in physics - from classical dynamics, to quantum mechanics, to general relativity - makes assumptions about the structure of spacetime. Among the most common assumptions are that spacetime is:

*

*uncountable (in the sense that the set of points that we identify as "spacetime" has cardinality greater than that of the natural numbers)

*everywhere dense (in the sense that for any two points $a\ne b$ there is a third point $c$ such that the distance from $a$ to $c$ is less than the distance from $a$ to $b$)

*smooth (in the sense that every geodesic is a differentiable curve)

*[almost] everywhere continuous (in the sense that spacetime is simply connected outside of singularities)

The application of analysis, probability, and topology to physics relies on these assumptions; in almost every case, spacetime is treated as a subspace of real or complex $n$-space (usually $\Bbb{R}^4$).
Now I could ask "but what if spacetime doesn't work that way?" - but that dead horse has already been beaten enough. Instead, I have a much more practical question:
Why should spacetime have the structure of a smooth, connected, topological, vector, etc. space?
That is, what observations or experiments, if any, suggest the continuity/uncountability/density/and-so-on of spacetime?

I've found this question particularly difficult to answer - chiefly because I've been unable to reconcile the infinite precision of mathematical statements with the practical limitations of physical measurement. For every case I can think of, the conclusion can be invalidated by the existence of a nonzero margin of error.
For example, we could say that a countable spacetime ought to have a nonzero probability of some event $\gamma$ occurring within a particular countable set $S$. We could perform measurements to show that the frequency of $\gamma$ within $S$ matches that predicted by the calculated probability - but there's some tolerance in our measurement that raises the possibility that the event only ever occurs sufficiently close to, but never at, a point in our set.
This all happens without regards to the details of the setup - which may very well require an infinite number of points and/or measurements, or any number of other impossibilities. Even so, the fact that any measurement will only indicate that an event has occurred within some minimal distance of a point in $S$ means that an uncountable spacetime could yield the same result. That is, if the probability of observing $\gamma$ in $S$ is $0$ - courtesy of the measure on $S$ being $0$ - we might still observe $\gamma$ sufficiently close to $S$, because the probability of $\gamma$ occuring within a nonzero distance of $S$ is nonzero.
Similar arguments can be constructed for the other assumed properties of spacetime. In all cases, it seems that the same results can be obtained in one hypothetical spacetime as can be in another as long as there is some tolerance for error in the measurement.

Edit:
In response to comments, and the recent edit of this question, I would like to address two things.
Firstly, I get the impression that this question is being interpreted as "why isn't spacetime discrete?" This seems to be based on an assumed dichotomy between "continuous" and "discrete" spacetime. This dichotomy is, however, almost entirely artificial. I listed the above four criteria individually because they are, for the most part, independent of one another. There are hypothetical spacetimes which are neither continuous (in the sense usually intended), nor discrete. For example, spacetime could be continuous without being smooth, uncountable without being dense, or dense without being continuous. Without some kind of experimental verification, the choice between any of these amounts to a matter of... well, choice.
Secondly, I am not asking for an explanation of the "true nature" of spacetime - as it stands, such a question would be meaningless. The issue I am trying to address is that vastly different spacetimes "look" the same below a certain (possibly infinite) "resolution". For instance, a spacetime which is uniformly "rough" at the sub-subatomic scale is indistinguishable from a uniformly smooth spacetime at the cosmic scale. This is the problem I am facing: how do we verify that our spacetime is the smooth one rather than the rough one (or uncountable rather than countable, etc.).
 A: The answer to this question is trivial. We don't know that spacetime is uncountable, continuous, or dense. These are not scientific questions, because we can't test non-discreteness. If we take the real number line as a model of time, or a smooth manifold as a model of spacetime, we're doing so because the model is convenient. Experiments have finite precision, so we can't test, for example, whether the ratio of two time intervals is an irrational number.

That is, what observations or experiments, if any, suggest the continuity/uncountability/density/and-so-on of spacetime?

None. This is not possible, even in principle.

The application of analysis, probability, and topology to physics relies on these assumptions; in almost every case, spacetime is treated as a subspace of real or complex n-space (usually ℝ4R4).

They depend on these assumptions for convenience, not of necessity.
A: We know that there is no such thing as a physical continuum. While we routinely approximate physical systems as a continuum, as soon as the dynamical degrees of freedom in what is supposed to be a continuum become physically relevant, we are forced to regularize this continuum or we'll end up with infinite results. In classical physics this problem is usually hidden, the physical degrees of freedom for small length scales can usually be isolated from degrees of freedom for longer length scales.
However, problems with the continuum do crop up in classical physics e.g. when we consider point masses or point charges. A well known problem in classical electrodynamics is to correctly describe the interaction of point charges with their own fields. This problem was solved recently using a regularization procedure. This problem and the way it was resolved clearly demonstrates what I said above: as soon as you have physical degrees of freedom that reside in the continuum in an essential way (such as point charges), the theory will break down and the only way to save the theory is to regularize these degrees of freedom.
In case of quantum physics, the problems with the continuum are immediately visible because all physical degrees of freedom at arbitrary small length scales are always relevant in a theory with interactions. Theories must always be regularized or else one will get infinite results.
We also know from quite general arguments based on quantum gravity that the number of degrees of freedom of a system with a finite volume is always finite. From ordinary quantum mechanics we know that a system with a finite volume with some upper limit on its energy content has a finite number of physically distinct states. If we invoke general relativity then the upper limit on the energy cannot be chosen arbitrarily high as the system will collapse into a black hole when the energy content exceeds a certain limit. Much more can be said about this, but the argument against a physical continuum are quite fundamental, they don't depend on the intricate details of the theories considered today.
Now, many people do think that the continuum does exist in some sense, they argue that the only way you can do calculus is by invoking the continuum. Since without calculus you can't do physics, it seems that the continuum does exist in some sense. But this assumes that the only way to set up calculus is via the convention approach involving the continuum. One can just as well replace calculus by discrete calculus where we have discrete derivatives and summations instead of ordinary derivatives and integrals. The results of ordinary calculus can be recovered by taking the limit of the step size to zero at the end of computations within discrete calculus.
Such limits are not so trivial because one has to replace continuous functions by properly regularized discrete functions. The continuum limit then becomes the same sort of elaborate regularization and renormalization procedure we are used to doing in physics.
A: As user268972 observes, a finite number of observations cannot, even in principle, empirically show the existence of an infinite structure. However, taken together, if we assume that spacetime is something in its own right, and if we also assume the invariance of physical law under Lorentz transformations (which is difficult not to assume) then we must assume an underlying spacetime continuum, because there is no finite representation of Lorentz group (sorry, I cannot remember who proved this, but it is at least intuitively obvious). We cannot therefore simply replace the spacetime continuum with some kind of discretised form of spacetime. The question is whether spacetime exists at all, or is it just a convenient mathematical concept which makes thinking easier. Both views have been expressed throughout history. Imv the most interesting part of this is that C20th science began to find answers rooted in empiricism and mathematics, not just the rhetoric of philosophers.
Leucippus, the creator of atomism, and his student, Democritus, postulated the notion of the void, which had no properties. In particular, place has no meaning in the void. Parmenides disputed, saying that something with no properties cannot be said to exist. For Aristotle the notion of position always exists in space (as reflected in the later atomism of Epicurus). But around 900 AD, a mediaeval encyclopedia of science and philosophy, written by a group of anonymous writers, the Ikhwân al-Safâ’ or "Brethren of Purity" contained the view “Space is a form abstracted from matter and exists only in consciousness.”
Newton followed Aristotle, and brought the mathematical structure of Absolute Space into physics, because he believed that it was required by the working of his laws. In special relativity Einstein replaced Absolute Space by defining coordinates in terms of measurement results, but then he spacetime, finding physics "unthinkable" without it. Ultimately, this was his underlying problem with quantum mechanics.
However, Sir Arthur Stanley Eddington was one of the first to endorse relativity precisely
because he understood its empirical basis. According to Eddington,
physical quantities are what we measure, “A physical quantity is defined by the series of
operations and calculations of which it is the result.” (1923, pre-quantum theory). This view was essentially the basis of Dirac's and von Neumann's approach to quantum mechanics, that it is formulated in terms of measurement results, not in terms of an underlying space or spacetime.
The idea gains substance in Feynman's approach to quantum electrodynamics, in which Feynman diagrams are conceived to model actual physical processes

“In Feynman’s theory the graph corresponding to a particular matrix element is
regarded, not merely as an aid to calculation, but as a picture of the physical process
which gives rise to that matrix element” - Freeman Dyson.

Mathematically, Feynman diagrams are graphs. The configuration of lines and vertices has meaning, the paper on which they are drawn does not. In particular, position has no role in Feynman rules. Thus Feynman diagrams mathematically represent the original notion of Atoms and the Void, due to Leucippus and Democritus.
I have given a complete discussion and mathematical treatment in my books, and in condensed, but mathematically rigorous, form in Mathematical Implications of Relationism, showing that the mathematical structure of modern physics does not depend on the assumption of substantive space or spacetime.
