The Falling Cat Problem main issue is that having no fulcrum points a cat cannot change its angular momentum
. If the angular momentum equals zero at the start, it remains zero.
We used to think that cats turned themselves round by swinging their tail in one direction leading the rest of their body to swing in the opposite direction, keeping the angular momentum at zero. Sounds reasonable, however the tail itself doesn't seem to be enough. As the footage of real-life experiments shows, cats move in a much more complex way. More so, even without a tail a cat will be able to turn itself round just fine.