... I did not dare connect the ground of my scope to the negative of the power supply, not knowing how it is related to the earth. The signal was present but very dirty.
You need to connect both or any information is worthless. Here is how I would go about this and all you need is a resistance meter.
First, confirm the earth from the wall plug is good. It should be good but I have seen cases where it was not connected just floating. Measure resistance between the ground in the outlet and a pipe or other grounded metal object and you should get a very low reading. (The house I am presently in was missing a connection and all the upstairs grounds were not actually connected to ground, just floating.)
With the washer disconnected from the mains and the board plugged in measure the resistance between the negative of the board and the earth-frame of the washer. Chances are they are connected together and that is your answer. If all readings between the board and the frame are in the many megaohm range then the board is floating. I think this is unlikely in a washer, but it is possible.
If the board has its own PSU connected to the mains then you need to be extremely careful while working on it.
Even then if it has a SMPS chances are the output negative is grounded. In this case I would cover or protect the input part so there is no chance of touching anything there by mistake.