What experiments at quantum or cosmic level might provide the strongest supporting evidence for or, conversely, potential for falsifying, the existence of dark matter or dark energy?
The ALPHA experiment at CERN is attempting to ascertain the gravitational properties of anti-hydrogen. Most physicists expect that anti-matter will fall (be attracted) in a gravitational potential generated by matter. If it turns out that anti-matter is repelled by ordinary matter, then this could explain dark energy and also solve the baryon asymmetry problem.
In this scenario the lack of evidence for anti-matter could be an artifact of our limited ability to detect it's signature outside of our local super cluster of galaxies (which would be devoid of anti-matter). Other super clusters could be made entirely of anti-matter (and repell super clusters made of ordinary matter) and the universe could be matter/anti-matter symmetric.
For this to be consistent with the large scale distribution of dark matter (connected filaments with galatic super clusters arrayed like beads on a string), it would be necessary for the dark matter partcle (whatever it is) to be it's own anti-particle and for dark matter to be attractive to both matter and anti-matter (of the ordinary type) under gravity.
As it would show if there is difference in Gravitational constant, and might thus open the door to new ideas about the mechanic of Gravity. If the gravitational constant is not a constant, then there is no need for dark matter and dark energy. I have a question about this issue; Gravitational constant, $G$, What if it is not Constant? and I have even tried to improve the idea further with this question; https://physics.stackexchange.com/questions/220050/le-sages-theory-of-gravitation-is-drag-the-only-reason-why-this-theory-fails
There is also MoND theories, "Modified Newtonian dynamics" which would falsify the dark Matter theories. It has shown some success explaining the dynamics of galaxies. One of its aspects is the Implications of Unruh-inertia to theories of gravity There is some resent papers published about these issues; ie. Testing Quantised inertia on Galactic scales