It's quite difficult to get a circuit to have a clean behavior at power-on and power-off: filtered reference voltages may take some time to stabilize, opamps don't work properly until they get the required supply voltage, microcontrollers have to boot, etc.
However, it is absolutely possible to make a bench power supply that starts up and shuts down cleanly, without exceeding the user set voltage. Some do that with an output relay that only connects the load when output voltage is stabilized, others do it by design... and others fail and make spikes at the output.
In my opinion, we should encourage manufacturers to fix this by not buying power supplies that spike at turn-on or turn-off.
Then, the most likely way to damage stuff is to set the power supply to a high voltage, forget about it, then the next day plug in a microcontroller, and turn it on.
Another way of destroying stuff occurs with analog supplies: when the potentiometer eventually dies, if it is a good design the output voltage will fall to zero. If it is a bad design, it will max out. There is no way to know without looking at the schematics. I've had this happen, it jumped from +15V to +40V when the pot died, and it fried all the opamps.
Does this apply to linear or switching power supplies?
Does this apply to power supplies with a seperate output switch?
If the supply powers up to the "off" state and you have to push the button to turn on the output after it has stabilized, that's one way to avoid the problem.
Is this caused by user error (is switching on a power supply with the load attached user error, or is there a seperate issue such as long leads)?
It's only user error if you know it's going to spike and turn it on anyway. Basically, don't buy a model that does it. A bit of search on forums should answer the question.
What I would call "user error" would be to set the voltage to, say, 10V and current to 20mA, then plug a LED into the output. The current limit only begins to work after the output capacitor has discharged into the LED, and at that point it is already fried. This is a case where you must manually ramp up the voltage. Another mistake would be to expect it will be stable with any load capacitance, then use a load with high transient current, which can cause overshoot.
I have a pair of Korad KA3005P, pretty cheap, and they behave well. The UI is nice except for the output relay, which has a quirky UI wich takes a long press on a button. It boots up in the ON state, which makes the relay a bit useless. However it is controllable with USB remote control, which is very useful.