I discovered that while my Siglent SDS1052DL oscilloscope has about a megohm of input impedance, the ground of each channel is actually common.
I accidentally connected a 12 volt, 25 amp power supply across through the ground of the scopes channels. The smoke came out of the ground clip on the oscilloscope probes. Both got hot enough to melt the insulation off the wire. The cable that leads out to the probe was not even warm to the touch. The rest of the probe appears fine.
I tested the oscilloscope against its 1 khz source that is on the front and both channels seem to work still. I am assuming whatever current path it took through the inside of the scope is actually very large in terms of conductor size, so the scope is probably fine.