How risky are 3rd party lenses?

They're going to test the hell out of that stuff!

So do all lens manufacturers. It's not like Sigma (or any other) is going to gamble when releasing a lens.

If your favourite 3rd party lens suddenly stops working

If a lens suddenly stops working, it's more likely due to some hardware failure of the lens instead of the lens becoming incompatible with the camera.

Is the lens manufacturer going to care

As far as I remember, Sigma and Tamron offer ways to update firmwares for lenses. Others might do the same. This should allow them to tweak the software of the lens to make it compatible with the tweaked software of the camera.

Other than that, if you are owning such lenses, you should be careful before you update any new firmware for your camera, because it has a (low) chance of making your lenses incompatible. Contact the lens manufacturer's support and ask for compatibility issues. If in doubt, don't do the upgrade on your gear. If you want to be sure, you could rent the same camera model with the new firmware and try it for yourself, although that sounds like overkill.

This risk does exist. My friend and I had first hand experience of this, but we made our best with it. He bought a used Sigma 70-300mm with a Macro setting at 300mm, that worked with the older Canon EOS models, and then suddenly on the Canon T2i, it stopped working. We called Sigma, and they said that sometimes, with newer camera firmware or after firmware updates, this tended to happen.

Luckily, I found out, the lens still worked with the T2i, only in the No-Flash, Manual and Sports Settings and none of the others. This worked perfectly for my needs, so I just bought the lens off him, and he moved on to a new one.

Also, you cannot say 1st party lenses will become unusable, just because of a bit of history. A few decades ago Canon did make the mount change and the old Canon FD lenses did stop working with the new ones. But such incidences are rare.

TL;DR: It depends on the age of the lens and whether or not you can update the firmware.

In reality, how common is it for a 3rd party lens to quit working due to an incompatibility? ... For that matter, what's the worst thing that can go wrong with a lens? Presumably a total communications failure would mean the camera doesn't know what lens it has and the autofocus doesn't work any more.

Bingo. Many lenses are still usable even if made incompatible electronically, but certain functions may cease to work properly, typically autofocus. Incompatibility doesn't mean the lens is rendered completely unusable, just that it may not communicate with the camera body correctly.

Is this a common problem or just a theoretical possibility?

If you're buying a new third-party lens that has some way of having its firmware upgraded, it's typically not that risky. The only thing you can't really be future-proofed against are physical mount changes (e.g., Canon FD → Canon EOS), in which case even first-party lenses are probably going to have issues.

Where the danger comes in is buying older used third-party lenses. There are reasons that lens may be going for super-cheap. Older third-party lenses had no way of upgrading the firmware at home, but typically had to be sent to service to physically have the chip on the lens replaced. And the third-party manufacturers would only offer to do this for a limited amount of time and may not have offered to do it for all lenses. If that chip-swap window was missed, there may be no way to get that lens 100% compatible again.

The risk of a 3rd party lens not working or to stop working with a camera is low, assuming the lens is made to work with that camera model in the first place. Third party lenses like Sigma and Tamron are major players in the industry, even if they are not as big as Canon or Nikon. Buying one of their lenses is safe. I tend to buy Canon lenses but I do have one Sigma lens, and I would buy others in the future.

My answer largely addresses your final point:

For that matter, what's the worst thing that can go wrong with a lens? Presumably a total communications failure would mean the camera doesn't know what lens it has and the autofocus doesn't work any more. It's not like the glass itself can suddenly stop refracting light! (Then again, my camera body refuses to shoot if it doesn't detect a lens attached, so many I'd be unable to shoot anything? No idea how aperture control works without electronic communication either...)

Lenses don't stop working. They stop having all of the features you expected.

You can use a 60 year old lens on a modern camera and with the right setup and manual know-how it'll still take great photos.

This risk everyone is talking about is only in terms of electronics and ease of operation. Its why I routinely am on here encouraging people even beginners to get well crafted lenses with aperture rings on them. It may not give me all the features a newer lens will but when a lens with no aperture ring no longer has a way to connect to a modern camera its a real pain to change the settings on. If you have a non-electric aperture ring this problem is solved.

It's the same reason the Nikon F2 is such a prized 35mm camera. The lack of electronics makes it incredibly long lasting and resilient. Electronics fail much quicker than mechanics.

A lens is a mechanical system of glass. It cannot fail unless you crack it, get significant haze or dust in it, etc. What will either fail or stop being compatible over time are electronics.

I'd also be very surprised if your camera can't shoot without a lens. Generally all you have to do is set everything to manual to shoot without a lens. There are a few cameras (Fuji for example) that require you to locate and change a setting first to allow the camera to perform this.

In the event you don't have an Aperture ring to change than you can either keep an old camera around just to change the Aperture or change it once to a decent aperture like 5.6 say and not worry about it much.

Since you asked for the theoretical worst possible things that could happen:

• An unexpected protocol incompatibility leads to a software crash that causes some microcontroller to operate itself or peripheral components in a fashion that is outside thermal and/or power consumption limits, damaging hardware and/or damaging data stored on connected media. For example, a program could get stuck in a loop around an instruction that has particular high power consumption, overheating the microcontroller and/or the power supply. Or, such a software crash could end up in behaviour that writes garbage to an SD card.

• If you had/have an interface design that leaves much of the protocol keeping to software (so called bit-banging), combined with hardware that needs strict discipline about the roles of combined input/output lines to be kept, there could likewise be problems leading to crashes or even damage: Two combined input/output lines that both think they should output make a short circuit.

• There might be undiscovered and undocumented features in every OEM lens that allow them to conform to demands only later equipment generations will make of them, regarding eg interface voltages or power management. This could be software commands ("you lens go slower now, you are blowing the power supply sky high otherwise!") never observed on older cameras. This could be hardware considerations ("Of course we built all the lenses so they will work on 2.7V supply and logic levels on future hardware, instead of the 3.3V we are using now! And don't worry about that metastability issue or this crosstalk issue, all the lenses can be relied to be in this range of signal rise times for their output lines...").

• There could be tight tolerances or material properties specified but unpublished for the mount parts in OEM lenses for the reason of not wearing the matching part out of tolerance. An out of tolerance design could work but cause increased wear. For example, an incompatible metal used could cause wear, corrosion, failure of lubrication...