Kinda Blender noob here. (v3, v4)
Wonder what my/the options are to do a none-linear scaling along one axe for the whole mesh/object. (Single plane-mesh distorted by a height map)
I have seen the falloff feature, but that did not do the job I had in mind for this case (not intended to do so either I figure).
Use a Lattice
If you are talking about proportional editing, it can be used if the non-linear scaling is non-linear in one of these ways:
It is intended precisely for that - any proportional non-linear transforms.
Just to be clear: using a lattice like Joseph suggests would be a lot more flexible and probably easier to use and I see no reason to use proportional editing instead, but you could if you wanted for some reason and the falloff modes matched what you need. I mean if we are talking about the options…