Thanks a lot for your suggestions!
Yes, i am able to apply a meter to the circuits, though i am not keen on getting shocked, i know already what it feels like
The high-voltage part in this oscilloscope is realized with two charge pumps as it seems, one gives -1400V, that is also where the intensity knob takes influence. I'd favor a defect in this part, as it is easy to reach
There's also a glow lamp in that part, and it glows continuously when power is on.
The other part is a charge pump that delivers +1600V (all numbers written on the boards). That part is nifty to reach and it would mean quite some disassembling work, including desoldering of wires.
So far all parts look sane from their appearance, and there is no buzzing sound or such. I'd also measure all voltages from the power-supply as well, all values are printed on the board as well. So i can step by step exclude components.
At least none of the caps in the first charge pump is short and they seem to charge/discharge, all diodes in the pumps (cascades) are okay as well.