What are all the ways in which a physical object or substance can appear to have colour, and how do they actually work?

There are lots of transparent and translucent substances that have color because of some kind of small scale inhomogeneity that absorbs light shorter than some wavelength. At least, this is what I picked up in several contexts such as photographic color filters, and the aging of paper. Therefore, if you want a longpass filter, which passes light of wavelength longer than some cutoff but absorbs light of shorter wavelength, there are many choices, with the cutoff in visible or ultraviolet or infrared. They’re tidy in the sense that the cutoff behaves in a nearly ideal way, including further and further from the cutoff wavelength.
There are other filters that select for colors according to other mechanisms, but they aren’t as numerous or tidy, not so many neat options. In the linked graph you can see lots of neat long-pass spectral curves, and a few not so neat other curves.
It’s kind of typical of things in nature. There are loads of substances that are yellow, orange, brown, red, or with a similar spectral curve but shifted into the UV or IR. Greens in nature are quite rare except for chlorophyll, and blues in nature are quite rare except for the sky.

Has anyone mentioned the whirly disc?

One of my favorite color mechanisms is the Christiansen Effect, named for (no joke) Christian Christiansen (1843-1917), a Danish physicist. He found that if you put one highly-transmitting material inside another highly-transmitting material that had the exact same refractive index at one wavelength, but with a large difference in dispersion between them, then you got what was effectively a color filter at the wavelength where their refractive indices coincide.

I’ve made several of these myself. You can easily do so with crushed glass immersed in a mixture of methyl salicylate (oil of wintergreen, which has a VERY high dispersion) and methanol. By varying the mix of methyl salicylate and methanol you can finely adjust the refractive index of the mix so that it coincides with the glass. The result is pretty striking. You transmit the light at the wavelength at which the refractive indices are equal, but the light scatters at other wavelengths, so you see a halo of colors around the directly transmitted light. Since temperature changes the refractive index (more in liquids than in solids) you can “tune” the color transmitted by changing the temperature of the mix.

Most explanations of how this effect works manage to botch the explanation. What’s really happened, as you can see by studying the mix under a microscope, is that you don’t get much of an effect where the light impacts the solid bits at normal incidence, or even pretty far from it. You get color effects at the edges of the particles, where the angle of incidence is very large, and you start to see prismatic effects. You want to have lots of these to get a lot of scatter, so it helps to have pretty small particles.

One of the more interesting results is not to use a solid phase immersed in a liquid one (like crushed glass in liquid), but to have two immiscible liquids. Then you can create the effect by agitating the mix, making lots of "bubbles’ of one phase in the other. They’re all pretty much spherical, so you get the color effect around the entire periphery. A mix that does this is glycerin in turpentine. It’s called a chromatic dispersion. Glycerin and turpentine at room temperature gives you a beautiful blue at the interface, even though neither of the liquids themselves have any color.

This is different from “invisibility”, where you immerse one phase in another, but the indices are very similar across the entire visible spectrum. T see the Christiansen effect you need the difference in index to change drastically with wavelength. So if you put pyrex glass in vegetable oil, or Teflon tape in ethanol, or those plastic water beads in water you’re not going to see any colors – the refractive index matches too well.

This is one of those things that would have made H.G. Wells’ Invisible Man even more unlikely. It’s not enough to simply match refractive index (as Wells’ guy did) – you have to match them at all wavelengths. Otherwise, your invisible man would be surrounded by a rainbow glow of color.

D’oh, I meant to write “transmitted”, there.

Cherenkov radiation could be a distinct entry on the list.

For some values of “physical object or substance”. And if we’re counting that, then there are a bunch of other processes to consider, too.

Aside…

One of my favorite plants in my garden are the spiderworts or tradescantia.
They are blue & violet; I have three shades.
They respond to ionizing radiation in near realtime.
They are edible, mucusy slimy, tricots, and are perennial here in Chicago.

Seems no more or less object-like than the blue sky mentioned in the OP.

Point. So let’s include some other processes, then. Synchrotron radiation, anyone?

Free-electron lasers! Electrons not wiggling the way you want them to? Just grab them by the neck and force them to wiggle.