Regulated power (voltage) supply operation:
To start from the big, more simplistic perspective, the job of the voltage regulators, both switching and linear is to act as ideal voltage source. That is to provide constant voltage even in case of varying load and/or its own supply.
Usually it is being achieved by using feedback loop. In such setting output voltage is being sensed and in case it drops below set value, something is done to provide more current to output which should result in output voltage going back to set value (and reverse if voltage is above set value). This "something" in linear regulators is to make pass element *1) to conduct more current from input to output by adjusting base current or gate voltage. In switching regulator usually "something" is to adjust frequency and/ or duty cycle to achieve the same goal. So in sum both linear and switching regulators main job is to reduce output voltage variations.
Now, nothing in life is perfect, and both realizations of the same goal have (severe) limitations. There is a lot of factors to to take into account (line, load regulation, regulation speed, stability, output noise, operating input/output voltage/current range and many many more) but for sake of (over)simplification linear regulators are better at providing ripple-less output then switched at the cost of the efficiency (this is beacouse switched regulator introduce its own ripple, but in turn are more efficient and can do things that linear regulators cannot do - such as stepping up voltage).
For the case from question:
A) In this application one really need good, regulated power source, as 50Hz/60Hz (100Hz/120Hz) ripple from mains are audible (so called power line hum). Also beacouse linked amplifiers trade supply variation immunity for simplicity.
B) LM317, from its DS, has 80dB *2) typical ripple rejection at 120Hz (power line x2). That is if you have 1V pk-pk ripple at the input your output ripple should be 0.1mV (10k times attenuation). Do not quote me on that (as there are a lot of factors to take care of) but it look like this should be more then enough for this application.
C) The switching regulator/power supply may be good enough providing that it does reject 100Hz/120Hz very well (80dB as in case of LM317 would be nice). Even as it produce more ripple (finding one with less then 5mV output self-ripple is difficult), if those are above 20kHz (and for most of the switching supply it is the case) you do not have to worry about it as those ripple would be outside frequency range that human ear can hear.
BTW you can look at linear regulators as "complicated voltage dividers"), which in fact is quite good analogy (as the pass element could be treated as "adjustable" resistance). Note, however, this "complication" gives you 80dB of ripple rejection :)
*1) pass element - usually it is transistor, BJT or MOSFET, connected between regulator input and output. The feedback loop will adjust it towards more "open" or "close" state such this element will pass more or less current in order to maintain output voltage.
*2) you need to design it in correctly, that is provide sufficient decoupling caps, make sure that it will work with appropriate drop to maintain regulation etc. Documentation is your friend.