Looks like mains pickup to me. Could be caused by the physical layout which is not good or high impedance inputs.
Just to be sure I understood you, you think there is a second wave superimposed to the pure sine wave and that second wave is caused by the mains AC line, probably caused by the cheap voltage supply I'm using.
I believe there are two ways to verify that: either I check the wave output using the exact same frequency of my mains which should produce an (almost) pure sine wave (two sine waves superimposed) or I can try a different, more stable, voltage supply (I've a stabilized 12V led transformer around).
I believe method 2 is going to be more reliable due to the micro oscillations in mains frequency to follow power requirements: do you agree/is my reasoning correct?
UPDATE: I just noticed I've already applied method 1 because I'm European and one of the frequencies I've diagrammed matches our mains at 230V 50Hz. :(
You are additionally mentioning a bad layout on the breadboard: can you explain what am I doing wrong so I can try to understand/learn/fix?
Thanks a million Mike, I'm learning a lot!