Common Fiat 500 failures + ones I've seen my friends experience
Multiair system; Costly replacement, surprisingly common. Can cause running issues such as reduced power and misfire Can also cause temporary failure of TC, ABS and Electromechanjcal steering systems.
Power steering Pump failure, caused by previous owner dry steering. Intermediate replacement cost
ECU failure, Oil warning light comes on, sometimes causes car to fail to start, car requires a new ECU to be programmed, which can be expensive.
Spontanious Power Loss, a famous issue was featured on watchdog, cars would lose power and be unable to climb hills. Software fix was deployed however
Oil leak fires, oil leaking from faulty gaskets and burning off the exhaust manifold, causing a fire.
Problems I've observed from friend's 500s
Failure to start: Low compression, engine turned iver extremely quickly. They scrapped the car as it was only worth £1500 and was uneconomical to repair
Auto transmission burning it's fuild; I was told this was a "cheap fix" but not sure exactly how much
It seems everyone who owns these fiats think they are reliable, however I would beg to differ. Although they could just be poorly looked after, since the people who drive them are not exactly Bruce McLaren.