Why does my single-supply non-inverting amplifier produce a wierd shape (depending on earphones)?
This is mostly the 'Impedance Match' issue. Every amplifier circuit is designed for a specific load impedance and can deliver the maximum power to that load. So by changing the load, you may not be able to deliver max power to the load, and also the output may contain harmonics (the distortion that you are seeing in the waveform). Take a look at this link:
Impedance Matching
It may help you to solve the issue.