Additionally to the thick, single layer grounded aluminium-shield which was already inside my preen, I dersperately added 216 Layers of aluminium-foil like discribed in the initial post by Xavier. This was just the maximum I could snuggly fit in between OLED-Module and Preen-Mainboard. These layers are not connected to anything and remove ANY(!) spurs I still had in the output.
I can measure down to -120dBFS and there is plainly nothing left. So, to me it seems like all spurs I had were related to the OLED-Module. It also seems to be just the magnetic field produced by the Buck-converter (PSU) on the module. Most probably (but I don't have a need to give this a test-run for now) ferromagnetic material (multiple thin iron-sheets) would be better suggestion for a shield against the magnetic field than using aluminium.
Stefan