How fast does the Earth's inner-core spin?
Roughly once per day, but a tiny bit faster. As the website for Columbia University's Lamont-Doherty Earth Observatory says:


  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...


Off the top of my head, I would have doubted that the core had a significantly different rotation rate than the mantle and lithosphere, because it would create different mixing layers in the liquid outer core, resulting in tangles in the Earth's magnetic field, similar to what happens in the Sun.