Does the Inner Core Change Shape?
And if it does, how do we tell and does it matter? The Inner Core is solid, the Outer core is liquid. Both are thought to be largely iron. The Core is responsible for the Earth's magnetic field, without which life on Earth would not be possible. And it is studied by using earthquakes, specifically those in the South Sandwich Islands and recorded in Northern Canada, passing through the earth's core on their way.
There is no shortage of South Sandwich earthquakes and many are very similar. Over the years their signature in Northern Canada has shown differences. The differences have been attributed to the Inner Core rotating at different speeds and lately the differences have been thought to be because of changes to the cores shape.
The above is a very superficial reading of the research. If you want to know more there is the BBC ARTICLE which is based on this SCIENTIFIC PAPER. The diagram below might help to understand the research.