Is there any evidence that supports or disproves the Oscillating Universe theory? What kind of observations could provide this evidence?
The main obstacle with the Oscillating Universe model is the pesky second law of thermodynamics — that entropy always increases within an isolated system, and never the reverse. Thus, as time progresses, entropy increases, until it (as far as we know) reaches its maximum — the heat death or Big Rip.
Consider the Big Bang, when entropy was at its minimum, or at least incredibly low. As the Universe evolved, entropy increased, as dictated by the second law. If the Big Crunch were to happen, we'd see a decrease in entropy as conditions in the Universe recede back to how they were at $t = 0$. As we've stated, the Universe's entropy can never decrease.
But say we ignore the whole "Big Crunch problem" and instead focus on the cycles. As stated before, the entropy always increases with time. I'll let Baum & Frampton (2006) explain the problem:
...one principal obstacle is the second law of thermodynamics which dictates that the entropy increases from cycle to cycle. If the cycles thereby become longer, extrapolation into the past will lead back to an initial singularity again, thus removing the motivation to consider an oscillatory universe in the first place.
As they described, the entropy in the Universe would increase with each cycle. At this point we realize that each cycle in the Universe's history would be different — and if we extrapolated back into the past, we would find that at some point, entropy must have been at its minimum (the Big Bang!) The fact that we would be back to the Big Bang would remove any need for an Oscillating Universe hypothesis — which tries to explain what preceded the Big Bang.
That problem led many astrophysicists to abandon the Oscillating Universe hypothesis. But nonetheless, the idea that the Universe is reborn is quite attractive, and we still know very little about cosmology — including dark energy, which we think has a vital role in the Universe. Some cosmologists continue to research the Oscillating Universe model in hopes that, as we learn more about dark energy and the Universe, we may be able to avoid the problems with the second law of thermodynamics.
The 2nd law of thermodynamics is not violated. If heat death is reached, and all energy reaches a minimal value distributed through space; you are no longer within the definitions of a closed system. There are no longer any thermodynamic state variables, nor is there a boundary. This is often overlooked. Oscillating Universe theories are still very possible.
The evidence suggests, strongly, that the expansion of the universe is accelerating. If an oscillating universe theory ruled that out then this would, in turn, dismiss this theory. As it is, it does strongly indicate that the idea that gravitational attraction would eventually slow the expansion of the universe and then send the expansion into reverse, is a false one.
But, you really need to be more specific in your question: which theory of an oscillating universe is it that you are asking about?