2002 Fiat Stilo shakes when accelerating

ok on most fiats they have a balance weight on the shaft but this may not be the case have you checked the tyres for any blebs or out of shape tyres also tracking of the car may cure the problem to if the wheels arnt running true

Its going to be a dodgy Ignition Coil mate. I have a Stilo Abarth too and they love to eat ignition coils! Get it pluged in and you should get a fault like misfire cylinder 1 - 5. Its dead easy to replace you just need to pop the engine cover off and use an allan key to remove the ignition coil over whatever cylinder has a fault.