The Earth's rotation is not stable, causing the need for leap seconds. How precisely is its instability known from moment to moment?
The way to measure it is probably to look at stars with a telescope, noting the time and telescope orientation, though the details of how to do it seem tricky. Maybe quasars and radio telescopes.
Inversely, how do astronomers keep their telescopes stably pointed at the same location in the sky for long exposure photographs if the earth to which their telescopes are anchored rotates unstably beneath them?
Sit down, then stand up. Your movement altered the Earth's rotational moment of inertia and consequently its rate of rotation, which in principle could be detected, then maybe the measurement deconvolved to recover your movement. Anything that involves movement of mass, for example air molecules vibrating in speech, affects the Earth's rotation. We can imagine a sophisticated surveillance system that monitors actions on Earth by seemingly absurdly observing distant quasars to extremely high precision.
Less ridiculously, the unstable rotation of the Earth could be a source of entropy for a random number generator. There used to be a random number generator powered by a Lava Lamp. The Earth is the ultimate Lava Lamp: variations in rotation are caused by movement of magma within the mantle or core. How many bits per second of entropy can the Earth's rotation produce?