I believe that it's common knowledge that Max-Q is the point in which a rocket is undergoing the maximum dynamic stress during a launch and ascent. But, how often have rockets actually been destroyed or failed at this point (or close to it) during launch due to Max-Q stresses? I've seen videos of many launches which fail with rapid disassembly before this point, but can't really recall that Max-Q is a statistically large factor in rocket launch failures. Am I mistaken, or have rocket designs evolved such that Max-Q is a known factor that's been successfully planned into rocket design and manufacture?
Rockets usually fail early in flight, within a minute of liftoff, or in the vacuum stage since that is hard to simulate on Earth. You can see a list of rocket failures here. The structure is one of the most reliable rocket parts, the failures are mainly related to fuel pumping and bad design decisions.
Both of the shuttle flights that suffered fatal accidents encountered significant wind shear events that took place very close to max q. I'm not aware that this was directly linked to the accidents but there were people involved in the STS-107 accident investigation who were very suspicious that this was a coincidence.
Max q was at 59 seconds. Max vehicle response to wind shear was at 61.724 seconds.
Max q was at ~58 seconds (I'm reading off a graph). The wind shear started at 57 seconds.
I just read of another incident, a Proton launch in Feb 1969, where the "newly designed" payload fairing collapsed at max q. (Soviet Robots in the Solar System, Huntress & Marov, p. 195).
The uncrewed Mercury-Atlas 1 flight suffered a catastrophic failure due to aerodynamic loads at or near max-Q, and the launcher's structure was beefed up for future flights:
[NASA's Owen Maynard] stated in an oral history interview that his post-flight calculations showed the skin of the launch vehicle just below the spacecraft would have buckled due to the combined drag, acceleration, and bending loads which exceeded the resisting tensile stress in the skin provided by internal pressure. Maynard recalled that "The problem of mating the Mercury capsule to the Atlas was far from being properly resolved at the time of MA-1." Based on that finding, the NASA specified that future Mercury-Atlas launch vehicles add doublers to the skin structure in that area, and that future launch trajectories be shallowed to reduce pitch angle rate, to reduce the bending stress on the launch vehicle.