How isotropic is the Earth's inner core boundary?
Some recent geophysical observations by Dai et al. (2012) (1) showed that compressional seismic waves reflected off the inner core boundary (ICB) exhibit significant variations in:


travel time, of the order of -2 to 2.5 seconds.
amplitude, by up to a factor greater than 4 (up to a factor of 6 oberved by Li et al. 2014 (2)).


These observations, according to Dai et al. (2012) (1) suggest that the ICB is irregular, with possible height differences of up to 14 km, across a distance of 6 km, and 4-8 km over a distance of 2-4 km.

Dai et al. (2012) (1) suggest that these variations are indicative of thermochemical deformation of the ICB. An alternative explanation presented by Li et al. (2014) (2), after numerical analyses of seismic traces, taking into account the effects of shallower structures, suggest that the ICB being a several-kilometre-thick 'mushy' layer indicative of a gradual transition between the inner and outer core.

The actual nature of the ICB is not completely understood, a recent theory by Vamos and Suciu (2014) (3) suggest that the anisotropy of seismic observations could be a result of a de-centred core due to interactions with the flow and the magnetic field inside the outer core. 

Observations by Wen (2006) (4) found that seismic traces from the same location taken at different times posessed different return times in the space of ~10 years, suggesting that deformations (as later described by Dai et al (1)) occur on a differentially rotating inner core, alternatively, Wen (4) suggests that the size of the inner core changes with time.

References

(1) Dai et al. 2012, Irregular topography at the Earth’s inner core boundary, PNAS

(2) Li et al. 2014, Notes on the variability of reflected inner core phases, Earthquake Science

(3) Vamos and Suciu 2014, Geophysical implications of a decentered inner core, arXiv.org

(4) Wen 2006, Localized Temporal Change of the Earth's Inner Core Boundary, Science