What is the most direct way of observation of quantization of angular momentum?
Rotational spectroscopy seems like a pretty obvious demonstration of quantised angular momentum. If you look at the microwave absorption spectrum of a diatomic molecule (with a non-zero dipole moment) you'll find equally spaced absorption lines. These arise because the angular momentum can only change in integral jumps of $\hbar$.
The Stern-Gerlach experiment also demonstrates (quite fantastically) the quantization of angular momentum.
The full details are in the Wikipedia page linked, but the essential principle of the experiment is this -
They sent a beam of silver atoms through an inhomogeneous magnetic field. Silver atoms have one unpaired electron, so the net angular momentum of the atom will be the spin angular momentum of the electron.
Classically you expect that the spin of the electron can be oriented along any random direction, therefore you'll see a spread of particles when they hit the photographic plate. But what they actually observed was two distinct beams of silver atoms. This was interpreted to mean that the angular momentum of an electron can have only two values: $\hbar/2$ and $-\hbar/2$.