Do electrons in an atom revolve around the the nucleus clockwise or counterclockwise?
Is there any rule to determine?
Lets make this clear:
Like other particles it can be also described as wave (wave–particle duality).
It may be a point particle, but it possibly has finite, but very small radius.
Electrons do move around nuclei. Corpuscular nature of electron completely excludes the possibility that electrons in atoms don't move. They also may or may not have true orbits depending on which interpretation of quantum mechanics is really "true". So, they move and are very fast; their velocities can be even relativistic (see this answer) and in such cases even Schrödinger equation doesn't describe atoms properly.
As Ivan pointed out in comments, orbits of electrons in atom, which wasn't disturbed in any way, can't be seen, which is true for any system (observer effect). Still even single electron during ionisation of an atom was observed using attosecond laser; a film of movement of electron cloud was created (see also).
As for OP's question "clockwise" or otherwise is relative to observer and therefore there's no point in asking that. Circular motion in original Bohr model was only an approximation and electrons move in unpredictable, chaotic way. Even on short timescales single atoms can be considered spherically symmetrical - they don't have left side or up side.