Can we easily observe a violation of classical mechanics? One might think that in order to observe a violation of classical mechanics, we have to have the high enough technology to observe the interference of individual photons or something like that. I think that if you're good enough at mathematics, you can directly derive a contradiction to classical mechanics from observations all the time.
I think one way to derive a contradiction would be from the fact that we observe that the presence of a macroscopic linear crack in the glass doesn't reduce its strength to zero. One possible conclusion from this is that glass has an infinite theoretical strength because the shorter the crack is, the more tension you need to apply in order for the crack to start propagating at the speed of sound in glass and make it appear to instantly fracture. We also know that glass doesn't have an infinite strength because we observe it to have a finite shear modulus and its strength can't be that much larger than its shear modulus. Under the assumption of classical mechanics, the only way for the strength of glass to not be reduced to zero by the presence of a crack when it has a finite theoretical strength is if it's composed of tiny particles. From that we can derive the contradiction that glass is not a stable substance because atoms are not stable according to classical mechanics, and as it slowly releases more energy, it can be compressed even denser. Yet we observe that that's not the case. 
My question is: Do these observations I describe really prove that the universe doesn't follow classical mechanics?
 A: violations of classical physics that are readily observable today include:
1) line spectra of excited atoms
2) the structure of the periodic table
3) lasers 
4) the shape of the black-body spectrum
5) alpha-particle emission
6) time dilation at very high velocities
7) energy release by fission and fusion
8) scanning electron microscopy, and
9) LED's.
This is a partial list. I invite the professionals here to add to it. 
Your assertions about the strength of glasses are incorrect; no quantum or nonclassical effects need be invoked in order to furnish an accurate accounting of their behavior. 
A: Yes. Violation of classical mechanics is easily observed with modern experiments. A single example would be the Stern-Gerlach experiment, or even simpler the observation of double slit interference between electrons, and even Bucky Balls.
I want to make a point about your mode of thinking, though. Deriving “contradictions” between paradigms is not the correct way to go about things on Physics. In fact, it’s a trivial derivation by definition. The universe tells us what is true, through experiment. If our current model predicted it, great. If not, then we are missing something. But again. This is Physics, not math. We don’t “derive contradictions”. Rather, experiment carries most of that weight. 
