What is the evidence for the super-rotation of Earth's inner core? What are the geophysical observations that support (or contradict) the hypothesis that the Earth's inner core rotates at a faster rate than the Earth's mantle?
Summary of Answers:
1) Studies of seismic ‘body-waves’ travelling through the Earth’s inner core indicate the inner core is rotating 0.2 to 3 degrees faster than the mantle.  These studies are sensitive to heterogeneities in the inner core’s structure.
2) Analysis of the free-oscillations of the Earth following great earthquakes are insensitive to local structure indicate that the differential rotation rate has been within 0.2 degrees over the past 20 years and that the inner core is probably gravitationally locked to the Earth’s mantle.
3) Comparisons of seismic ‘body-waves’ travelling through the Earth’s inner core from  earthquake doublets (a pair of earthquakes that occur at the same place but different times,)  should also be insensitive to local structure (because the ray-paths would be the same) and can be used to investigate changes over relatively short intervals of time.  These studies indicate that the mean rate of rotation of the inner core has been 0.3 to 0.5 degrees per year faster than the mantle, but that the rate has also had decadal fluctuation about the mean of 1 degree per year. 
 A: Here are some supporting evidence, taken from here.

The inner core rotates in the same direction as the Earth and slightly faster, completing its once-a-day rotation about two-thirds of a second faster than the entire Earth. Over the past 100 years that extra speed has gained the core a quarter-turn on the planet as a whole, the scientists found. Such motion is remarkably fast for geological movements -- some 100,000 times faster than the drift of continents, they noted. The scientists made their finding by measuring changes in the speed of earthquake-generated seismic waves that pass through the inner core.
The Earth and the core are rotating on the same spin axis, but because the inner core rotates just a bit faster than the planet as a whole, the "fast axis" through the core moves eastward. Over the years, it traces a circular path around the north pole and moves to different positions relative to the Earth's mantle and crust. This basic feature allowed the Lamont scientists to make their discovery.
Dr. Song and Dr. Richards studied seismic waves from 38 earthquakes that occurred between 1967 and 1995 near the South Sandwich Islands at the bottom of the globe. They measured the speed of waves that traveled up through the inner core to receiving seismographs in Alaska at the top of the globe and found that the waves arrived about 0.3 seconds sooner in the 1990s than they did in the 1960s.
The speed of the waves steadily increased over the period because the inner core's "fast axis" -- the fastest possible route through the inner core -- was gradually becoming more closely aligned with the actual pathway traveled by the waves between South Sandwich and Alaska, the scientists said. The change in the waves' speed showed that the fast axis was in motion relative to the Earth, proving that the core is spinning faster than the Earth, they said.

Also, you can read the following 1, 2 and 3.
