If you had a short in place of the potentiometer, you should have read 1.25 volts (or so) at the output of the LM317, not 12.5 volts.
When you changed your meter to read current, presumably moving the red lead to the "10 Amp" socket, the meter would have very low resistance. When you then connected the meter between the LM317 output and ground, a very large current would flow, apparently causing the LM317 to explode.
If I recall correctly, the LM317 is spec'd to deliver 1 amp, with due regard to voltage drop, power dissipation, heatsinks and phase of the moon - that only states the current that it can safely carry under suitable conditions - it DOES NOT limit the current to that value!
The current measurement function of a multimeter should only be used with the meter connected in series with a load whose current drain you wish to measure. It must NEVER be connected directly across a power supply.
Is it correct to assume that an ideal (theoretical) PSU for audio applications should produce a constant voltage regardless of load variations (i.e. it is a voltage source)?
Yes. An ideal power supply for any application should be an ideal voltage source, which has a constant voltage.
In practice, what are acceptable levels of supply voltage variation due to transients in an audio application (in percentage of Vs or mV)?
This is dependent on your application. You have to evaluate your desired noise/distortion, the power supply rejection of the audio components you are using, and the way the circuit is constructed. 0.1% power supply variation translates to a -60dB noise floor, which might be sufficient.
What is the correct approach to reduce these V swings? Should I place a capacitor on the INPUT or OUTPUT of the regulator? Is there a rule of thumb/calculation for the required capacitance? Are electrolytic capacitors OK, or should I use polyester or tantalum caps (i.e. something with a lower ESR)?
Probably both. You should have both bulk capacitance on the output and low-ESR decoupling caps in close proximity to all active chips (op-amps, ADCs, DACs, etc). And some more capacitance on the input certainly wouldn't hurt.
Typically, you might use large electrolytics for bulk capacitance, and low-ESR ceramics for faster decoupling. Again, how much you need depends on the magnitude and characteristics of the wiggles on the supply rail. Also, carefully read the datasheet of the regulator and make sure you are within its comfortable operating region: the regulator has a response speed and current limits, as well as input ripple rejection specs.
Best Answer
The output voltage is determined not by the ratio of R1 to R2. It's given by the following equation:
\$ V_{OUT} = 1.25 \left(1+\frac{R_1}{R_2}\right) + I_{ADJ}R_2\$
For ordinary purposes, the \$I_{ADJ}R_2\$ term can be discarded, because \$I_{ADJ}\$ is on the order of \$100\mbox{ }{\mu}A\$.
You've multiplied your resistors by 10, so this error term will also be multiplied by 10, going from 33 mV to 330 mV, or 0.33 V.