Why do methane molecules have 15 degrees of freedom? I've just been learning a little thermal chemistry including the equipartition theorem. As part of this, my textbook discusses how to figure out the degrees of freedom for different chemicals.

The answer to one of the exercises says methane has 15 degrees of freedom: 3 for translation, 3 for rotations and 9 for vibrations. I am confused by two of these.

Firstly the text says that diatomic molecules like oxygen gas only have 2 rotational degrees of freedom, because they can't rotate about their axis of symmetry. However methane has an axis of symmetry, so why does it have 3 rotational degrees?

Secondly I don't understand where all the vibration degrees of freedom come from. The molecule has 4 bonds, so wouldn't each bond have one degree for kinetic energy and one for potential energy, for a total of 8 degrees, not 9?
15 is there because you have 5 atoms which can move independently, and we live in a 3D space, and $3\ \times\ 5\ =\ 15$.

3 of these are translational degrees of freedom, which you seem to agree with, so I'll just leave it at that.

3 are for rotations, because the methane molecule is not linear. Axes of symmetry are irrelevant: indeed, you may rotate methane around the axis of symmetry as well as around any other axis. Being or not being diatomic is hardly relevant either, except for the fact that all diatomic molecules are necessarily linear; however, the converse is not true.

Subtracting 3 and 3 from 15, we get 9, and that's how many vibrational degrees of freedom we have, because they can't be anything else. To visualize them is another story, and a great deal more difficult. Counting bonds doesn't help much. The separation of degrees into kinetic and potential is not a thing at all. You don't have to do that anyway. Write $15\ -\ 3\ -\ 3\ =\ 9$ and call it a day.