There is a hidden trap for new players in the "social contract" of most DC power supplies: the current limit applies before the output capacitor. This means if you set your power supply to 10V 1mA and connect an LED directly to the output (or connect both outputs to ground), the power supply will not pump more than 1mA into its output capacitor but the output capacitor will happily dump far more than 1mA into the LED until its voltage has fallen to steady state. This kills the LED (or causes a spark) because the short circuit current of the capacitor is large. Intro EE classes often buy fresh LEDs for their first lab class because it is expected that many students will learn the hard way, even if the instructions try to warn. You are not alone in this, it's a universal experience.
I have noticed extra caveats that come with cheap power supplies. A common one is that if you do not turn off their output before cutting power, they dump their intermediate voltage caps directly into the output (in my case, this was 42V for about 20ms). I have also seen a protection function that when engaged does not go to zero but rather outputs -1.2V and -20mA on a supply that was supposed to be positive only. The bigger point is: cheap power supplies do weird things. I have not seen the behavior of periodically shorting the negative terminal to ground, I would typically expect the terminal to be isolated, but I would not be surprised to see such silly behavior from a cheap supply.
As you might expect, more expensive supplies tend to diverge from baseline in the opposite direction. Expensive power supplies from Keysight (or so) will sometimes support "downprogramming" -- intentionally draining their own output capacitor to reduce voltage quickly for waveform generation or to quickly enforce current limits. SMUs also typically don't have the output capacitor caveat. If you set them to 10V 1mA and short the output, the only surge will be the surge required to drain the parasitic capacitance of the wires. The 1mA limit will apply inside a microsecond rather than after many milliseconds. You pay a pretty penny for these features, though, so it is best to learn how to deal with regular power supplies first.
The oscilloscope is probably fine. They are usually constructed with probe ground directly connected to the chassis. You could likely dump enough current into your probe ground to make the wires glow red hot without damaging the actual signal processing hardware in the scope. The melted probe would need to be replaced, of course, but that would be the end of it.