Vibrating speedo needle

[powerandtorque]

Just to clarify what I meant, this is the cable driven setup from an earlier Phase 1:



And this is the later cable-less setup that's electronically driven (but still with a mechanical odometer) from a late Phase 1:

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The usual issue with the cable is that the grease inside dries up and causes the cable inner to no longer rotate smoothly, leading to the needle shake.

I don't know how bad 306's are for it as I've predominantly had later cars with the electronic odometer, but 205 GTi owners will be very familiar with the symptoms you've got as it was one of the shortcomings on the RHD models where Peugeot in their ultimate wisdom routed the speedo cable above the exhaust manifold, causing the speedo cables to dry up, shake and - eventually - fail altogether.