Oldskool linear transformer-diode-cap power supplies work great. However, ironically, the reasons behind their excellent short-term overload tolerance are the same reasons why you don't want to use them: they're huge, heavy, and therefore expensive.
In a linear power supply, if the transformer is sized for average power with a thermal cutout, it'll handle overloads just fine due to the huge thermal mass of copper. What "average power" is means different things in a PA amp for a nightclub subwoofer, or a living room amp.
The PA sub amp will most likely be used at full power almost continuously with signals that look like sine waves, so average power should be close to max power at onset of distortion. The living room amp will most likely be used at a few watts average, or 1/4 to 1/8 of peak power, so its power supply can be "cost-optimized".
However with switching power supplies you have two problems.
One, they regulate output voltage. That's nice, but it means the output capacitor has a constant voltage across it, so it can't be used to store energy.
With a SMPS, the high voltage capacitor on the primary provides energy storage, and since its voltage can be allowed to vary in quite a large range while keeping the output in regulation, it can store quite a lot of energy. The output cap only provides a low output impedance and smoothing at the SMPS' switching frequency, but at audio frequency, most of the AC current will come from the SMPS itself, and output voltage remains regulated.
That's nice, but it only works when the amp's supply current demand is between zero and the SMPS' current limit.
If the loudspeaker sends some back-EMF into the amp, then a class D amp will absorb it and store it in its power supply capacitor, just like a synchronous boost regulator. At this point the SMPS will probably go in standby until its output voltage returns to normal.
If the amp draws supply current above the SMPS's current limit, then it will either hiccup (very bad) or go in current limit mode (better) depending on the model. If it simply limits current then the output caps will provide some energy storage, and later recharge to the normal voltage.
But in case of a SMPS, since most of the current goes through it, it can't be downsized like an oldskool power supply. Either you want regulation at all times and it has to be rated for full power... or you don't want regulation, don't care if output voltage drops when the SMPS current limits, and pick a SMPS that actually current limits instead of hiccuping, and put big output capacitors... but in this case, I guess you should really do the math to check if the extra money spent on the big output caps could be better spent on a SMPS that's powerful enough to not need the big output caps, and rely on the high voltage input cap instead.
In the forums, competent class D designers really insist that a good SMPS is a vital part of a class D amp. A popular solution seems to be a SMPS with the same overload characteristics than the oldskool ones: basically, to spend a little more to beef up switching devices and other components that tend to fail very quickly when overloaded, so they can handle the full power for a few seconds... while saving money by not requiring it to handle full power continuously, so it can use cheaper alternatives to the components that don't fail quickly due to large thermal mass (ie, smaller heat sink, no fan, maybe skimp on capacitor ripple current a bit, etc). Basically, a good quality power supply for a class D audio amp would be able to handle the full power continuously with a larger heat sink and fan, and maybe more caps, but these are omitted.