With nothing connected to the scope, at 2V setting I get something like a tiny impulse train on top of the waveform. At 1V I get a perfect flat waveform. Is this normal?
It might actually have to do with the scope bandwidth:
When switching from 1 V/div to 2 V/div you should hear an internal relay clicking. This is because the internal attenuator is switched from 20 dB (x10) to 40 dB (x100).
On the other hand, the 20 dB bandwidth limiter is directly before the ADC, yet it is 1st order only hence not very effective.
At 2 V/div the signal is only one tenth of that at 1 V/div, hence S/N ratio is 20 dB worse. That's only a concern for injected noise, entering the signal path after the attenuator, as it might happen in very noisy environments. The scope itself is electrically quiet, see my test results below (well, at least an original SDS824X HD is).
I've tested the SDS824X HD with a zero signal to rule out external noise as a factor, i.e. just a 50 ohm terminator at the input. The only difference between 1 V/div and 2 V/div is the noise level more than doubles at the less sensitive setting. Yet it's a uniform white noise without spikes. Take note of the different vertical gain settings in the zoom window!