Statistically speaking, heating appliances, especially space heaters are the most dangerous electrical devices in the home from a fire perspective. Both directly and also in fires caused by the relatively large load causing poor connections and low quality extension cords and wiring to heat up.
Exactly what happened to one of mine, and not a cheap one. a 2kW oil-filled radiator in our outbuilding stopped working one Winter. It looked like the fuse (I think, or maybe the contacts to the fuse) in the plug had overheated to the point that it burned out the entire inside of the plug. Even the wall socket got scorched and cracked down the middle. It had been in continual use several winters, running months at a time on full, but still a bad and potentially serious failure.
Another failure I saw was from a mains socket where just a TV was plugged in, and had been for years in my parents house. That just started smoking one day when we were sat there. The cause: a sharply bent solid wire behind the plug. It must have been ever so slightly fractured at the bend, causing a slow heat build up at that spot, causing more damage from expansion... until it finally got thin enough to heat up from the TV load and melt the insulation.
The worst one happened just a few weeks ago in our current house, again thanks to someone else's dodgy wiring: I was on the way to bed one evening and as I walked past the hallway cupboard where the (really ancient) consumer unit is, I smelled 'hot electrics', just slightly. I could feel the heat when I opened the cupboard and the metal CU was hot to the touch. Inside, the moron who had wired it had shoved two wires together into one terminal, instead of using the empty one next to it. I guess they didn't make good contact and one had gradually overheated and burned away the insulation for 6 inches down the live wire and damaged other wires near it. It probably wasn't far off being a fire.