The most common faults are

Power cable break. Because of the way it swivels, the joint wears and you get intermittent or complete no-power.

Due to the wear and tear from use (and also wrapping cables around irons), the cable can break. Particularly where the cable enters the strain relief at each end. The sockets (apart from the type 2 ones) are pretty reliable. If the cable fails internally as a short, there might be sparking and a blown plug fuse

Thermal Fuse. Often the plug fuse (UK) will blow also. The symptoms are the unit not powering on at all. Quite often a faulty thermistor will cause the thermal fuse to blow. So if you only change the thermal fuse, it could blow again. To be safe, change the thermistor at the same time as the thermal fuse

The thermal fuse is on the back of the plate(s) and fails due to a broken element or a detached thermal fuse (depending on model). The thermal fuse only fails for no reason on the 3.1B model, because they speced it at 218degC rather than 227degC. On newer models (Eclipse onwards) there might be signs of life, even if the thermal fuse has tripped

Thermistor. Usually the unit powers on and beeps, but no heat in the plates, unless it has also caused the thermal fuse to fail, in which case see bullet point above

R8/R11 Resistor failure on mk4 and mk5 is VERY common. Easy to fix but can also cause heater element to fail as well. See this link for more details

Other common faults can be found at this link.