What is the chance that a normal household lamp with a lightbulb (LED in this case) left constantly on will cause a fire? I realise that in the case of correct installation and use this should not happen, but given certain events like power surges or material wear and tear this could eventually happen, right? What are the chances that it does? The specific case I'm wondering about is when a light is left on when no one is at home.
Under normal operating conditions, and assuming no failure in the device, it will be certified to not catch fire. (Though cheap imports may not be quite so rigorous on certification.)
However, if it is enclosed in such a way that heat can not escape, it will get hotter and hotter till it either fails or catches fire.
If the light has areas that are designed to be very hot, like a halogen bulb, then measures need to be taken to ensure flammable materials do not come into contact with it.
Also, socket/fixture it is plugged into must be rated to take at least that bulb wattage and size.
Bit hand wavy here...
Most LED lightbulbs are sealed and built either with flame-retardant plastic, ceramic-loaded high thermal conductivity plastic, or aluminium. All of those are quite stubborn in not catching fire.
Also, there is a fuse, by regulation, so it is unlikely any internal short would last long enough to heat up anything to flashpoint.
In the end, the tiny AC-DC switchmode converter inside a lightbulb isn't that different from a mains-USB charger, and likely to present the same safety issues. So if you are paranoid, switch everything off...
I would be more worried about ultra-cheap chinese products, though, because if they "cost-optimize" the fuse out, and then use hair-thin wires, haphazardly assembled, then your circuit breaker may not protect you in case of an internal short.
Some LED bulbs also have the LEDs at live mains voltage, so there is a risk if the cover falls off due to mishandling or dropping.
However, I think flammable materials in direct vicinity are more of a risk, like paper lampshades, carpeting, or the worst of all, a christmas tree: