As many have pointed out, the double-slit experiment is perhaps the canonical demonstration of superposition and a macroscopically visible demonstration of quantum mechanics. But the fundamental point of the double-slit experiment is to demonstrate that matter is wave-like. And to say that it is wave-like is to say that it is extended in space, periodic in some sense, and has the ability to interfere with itself. These concepts simply do not exist in classical theories which assume that matter is point-like, which is basically the diametrically opposite view.
If you set up a screen with two vertical slits, and shoot paintballs through the slits, as long as the slits are wide enough to accommodate the paintballs, you will see two vertical lines of paint on the target behind the slits. And this comports with the classical prediction using classical point particles.
If you create a seawall and make two small holes in it, and then start generating waves on one side of it, the waves will hit the wall, pass through the holes, and interfere with itself on the other side, creating a pattern of peaks and troughs that is far more complicated than the original wave. So classical theory can deal with waves just fine. It just doesn't admit that point particles are waves.
The difference between classical and quantum mechanics is that QM says that everything is a wave, and thus, has a wavelength. And thus, everything can, in principle, be made to interfere with itself. What this means in practice can be a bit subtle, but suffice it to say that this prediction is noticeably absent from classical theories.
Now, when we pass electrons through the double-slit experiment, an easy objection is to say that the interference happens because multiple electrons are passing through at the same time and interfering with each other. Thus, individual electrons are not really waves...the experimental artifact is due to collections of electrons acting as a wave together (just like ocean water hitting the seawall). However, this objection is defeated by passing electrons through the experiment one at a time. Classical theory cannot make a wave out of a single particle.
But what does it mean for a particle to pass through two slits as a wave? The way we detect the electron on the target is by, say, a glowing phospor. Since an electron can only hit one phosphor, what does it mean to say that it's wave-like? At this point, it's looking very particle-like, and it's easy to imagine that it only took one well-defined path through the experiment. The problem is which path did it take. The interference pattern does not show up with a single electron, because you can't make a pattern from a single data point. You need many electrons to build up a pattern. And what you see is that instead of just hitting the target in two well-defined lines, the electrons are spread out horizontally, but in clumps. How do they know to do that? If the electrons are taking a simple direct path through one slit or the other, why would they strike the back target in one of a dozen or more clusters? Where did those clusters come from? The QM answer is that the electron takes both paths simultaneously, thus causing it to interfere with itself, with the final point of impact determined by a probability distribution that varies in intensity across the target. And so far, nobody has come up with a more economical explanation than that.
Classical theory says that a charge like an electron orbiting another charge like a proton is accelerating and thus should be constantly emitting radiation and spiraling into each other. Classical theory tells us that our precisely localized electron point particles behaving according to Newtonian mechanics should cause all matter to collapse on itself in a blinding flash of light faster than you can say: "By Grabthar's Hammer!" The fact that it doesn't tells us that classical mechanics is missing something important. Quantum mechanics says that electrons don't spiral into atomic nuclei because they are not point particles, but waves. And bound states are much like waves in a musical instrument: the set of frequencies allowed is highly constrained by the size and geometry of the instrument (e.g., where it is fretted). And this conveniently tells us why electrons not only don't spiral into the nucleus, but also why they absorb and emit radiation at particular frequencies.
So when an electron is bound to a nucleus, where is it? Well, you could detect it at any number of locations, just like you could detect it in many places on the target screen in the double-slit experiment. But where you do detect it is not predicted by classical theory (which says that the place you detect it is crashing into the nucleus). Saying that the electron is definitely in one place or another before you've measured it is not a well-defined operation, because the electron is in a superposition of states. So you could say that pretty much all atomic theory since the description of the photoelectric effect is more or less a "proof" of quantum mechanics in general, and superposition in particular.