What allows us to infer the existence of dark energy? A recent Indian examination asked: how do we infer dark energy is there? It gave as options: rotation curves of galaxies, accelerated expansion of the universe, existence of microwave background radiation, and Hawking radiation from black holes.
I don't know the answer. So what observations allow us to infer the existence of dark energy?
 A: Hmm, this looks awfully like a homework question. Giving you the benefit of the doubt, the answer is (B). The universe is (to a good approximation) described by the FLRW metric. Dark energy causes the cosmological constant term to be greater than zero. This is responsible for the accelerated expansion.
A possible confusion is that (A) is probably caused by dark matter, which is different from dark energy.
Later:
David's edit of the question has left my original answer looking a bit odd, so even though I suspect Abhishek Pant isn't interested, let me answer the modified question, particularly since there is some evidence of dark energy that I didn't know about until I looked just now (not that the Physics Stack Exchange has become my personal blog, you understand :-).
We probably all know about the SN1a supernova data that shows distant supernova have been accelerating relative to us. This is the classic evidence for dark energy. See http://arxiv.org/abs/astro-ph/9805201 and http://arxiv.org/abs/astro-ph/9812133 for the original papers.
The second piece of evidence, again that we probably know about, is that the WMAP experiment shows the universe is flat to the accuracy we can measure, and since the visible and dark matter are only estimated to add up to 30% of the critical density the remaining 70% must be something else, and dark energy is the obvious candidate.
So far so good, but did you know that the Wigglez group have been measuring galaxy redshifts by estimating galactic distances from a study of the void in the galaxy distribution. They measure the same acceleration as the SN1a studies.
Finally there's something called the Sachs Wolfe effect that I don't really understand but is described at http://en.wikipedia.org/wiki/Sachs%E2%80%93Wolfe_effect.
