Hi Mossy
I had exactly the same fault with my Stilo 1.6. The ABS and ASR failure messages would appear immediately after starting the engine, then disappear and not be recored. It would happen most when the car was cold.
After FIAT tried suggesting it needed a new Steering Angle Sensor (a part which my car doesn't even have), I myself thought that it was the battery not having a big enough capacity to start the car and simultaneously supply enough current to the car's systems. I was told that the car checks ABS and ASR whilst the car is cranking, as it they are priority systems. Therefore I concluded that, if the battery was poor, these systems would not get enough power, hence not return the correct current/voltage, hence displaying a fault which would immediately disappear once the altenator punched the voltage and current up.
I got my dealer to replace the battery, and put the one from the JTD in (£14 extra for me to pay), and touch wood, the fault has never returned, and we've had some cold mornings here since then. Give it a go.
Hope that helps!
Stilo 1.6 Dynamic, 5dr, 2002, Crystal Green