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Where do they show up in classical mechanics?

Anywhere you have to solve the Laplace equation in spherical coordinates. For example with the gravitational field.

Vibrational modes of spherical stuff.

Fair point. We will fix the language up

It's a pretty minor point, but the pedant in me appreciates it nonetheless!

It's also true that since most people are not subjected to the horrors of learning about PDEs, the place where they may be exposed to spherical harmonics is in atomic orbitals from high school chemistry, so I could see where you were coming from.

I don't think that there is any need for "Technical nit:" and it looks a bit aggressive to me and without any substantiation.

Why not say your piece and be done at that point in time? You can respond to counter arguments as they arise in child comments.

Why not kick off with "Here at {wherever}, we find that ... crossing the streams is a really bad idea" or similar?