The report of super-rotation of the Earth's core was printed in Science in 2005 (paid subscription).
They studied 18 pairs of earthquakes which occurred in the same locations, but years apart. A free summary is available here: http://news.nationalgeographic.com/news/2005/08/0825_050825_earthcore.html
It is thought that the core takes about 900 years to rotate once compared to the surface of the Earth.
The progressive crystallisation of iron at the surface of the solid core would release latent heat, which drives convective motion in the mantle, which in turn is thought to produce Earth's magnetic field. The core itself probably can't produce a magnetic field, as it is above the Curie temperature of iron.
What is not clear is whether the rotation of the core is a cause or an effect. Convection would occur even without the rotation of the core. But there is a hint that perhaps the rotation of the core is driven
by the convection of the mantle.
More detailed study of earthquake waves is slowly producing a more detailed image of convection patterns within the mantle, by measuring arrival times of earthquake waves which arrive in the same direction as flows in the mantle, vs those that travel against it.