Is the spacetime continuous or discrete?
Or better, is the 4-dimensional spacetime of general-relativity discrete or continuous? What if we consider additional dimensions like string theory hypothesizes? Are those compact additional dimensions discrete or continuous? 

Are there experimental evidences of continuity/discreteness?

When particles move inside space do they occupy spacetime by little chunks?
What would imply if spacetime is discrete on continuous theories?

I've found little information on the web and books.

Probably my question is ill-posed and I apologize for this.


  is the 4-dimensional spacetime of general-relativity discrete or continuous?


In the usual definition of general relativity, spacetime is continuous. However, general relativity is a classical theory and does not take quantum effects into account. Such effects are expected to show up at very short distances, where your question is relevant.


  Are there experimental evidences of continuity/discreteness?


All the experimental evidence points to continuous space, down to the shortest distances at which we have been able to measure. We don't know what happens at shorter distances. We also do not have any direct experimental evidence that gravity is a quantum theory, with the same caveat.

On the other hand, we are quite confident that a complete theory of nature must include quantum gravity and not just classical gravity. And, we have an educated guess of the distance scale at which quantum effects should become measurable: this is the Planck length, roughly $10^{-33}$ cm. This is much much shorter than the shortest distance at which we can carry out experiments, so at least we are not surprised that we did not see any such effects so far.

Before proceeding, one more caveat. There is an interesting and quite recent astrophysical experiment that showed that Lorentz symmetry holds even below the Planck length. If Lorentz symmetry is broken, it generally means that photons with different energies will travel at different velocities. At the experiment, they managed to detect a pair of photons that were created at almost the same time but had very different energies. They reached the detector almost simultaneously, which means their velocities were similar. Because the photons travelled an enormous distance before reaching us, they must have had almost the same velocity.

So we know that at least Lorentz symmetry holds at very short distances, and it seems difficult to reconcile this experimental fact with a discrete spacetime. So at least naively it seems that this is evidence against discreteness.


  Is the spacetime continuous or discrete? 


At long distances spacetime can certainly be thought of as continuous. At short distances, the short answer is: we don't know.

String theory is the only consistent theory of quantum gravity we know of, where we can actually compute things with some confidence. (You will probably hear some opinions that contradict this statement, mentioning loop quantum gravity, causal sets, etc., which are not related to string theory, but what I said is the common view in the community of high-energy theorists.) String theory is giving us some strong hints that perhaps spacetime at short distances is not continuous or discrete, but something else that we don't understand yet.

So the situation is that even theoretically, without talking about actual experiments that check the theory, we don't know what spacetime is like at short distances. Perhaps this is why you don't see this question mentioned a lot. My personal guess is that spacetime at short distances is neither continuous nor discrete, but has a different nature that may require new mathematical tools to describe.


  Or better,  What if we consider additional dimensions like string theory hypothesizes? Are those compact additional dimensions discrete or continuous?


Adding extra dimensions does not change any of the above.
