Has a reduction in entropy ever been observed? On the whole, the entropy of the universe is always increasing. There are far more possible states of "high" entropy than there are of "low" entropy. The example I've seen most often is an egg$-$ there is only one way for an egg to be whole (the low-entropy state), but many possible high-entropy states.
Now, this is only a "statistical" law, however, and, as I understand it, it's not impossible (only incredibly unlikely) for a metaphorical egg to "unbreak".
My question, then, is: have we ever observed a natural example of a "spontaneous" reduction in entropy? Are there any physical processes which work to reduce entropy (and, if so, is there a more concrete way to define entropy? Obviously, a cracked egg is intuitively a higher entropy state than an "uncracked" egg... but what to we really mean by entropy, which is distinct from disorder)?
 A: 
My question, then, is: have we ever observed a natural example of a
"spontaneous" reduction in entropy?

Yes, of course, but as @Mark_Bell pointed out, it depends on if you are talking about the entropy of the system only (or the surroundings only) or the total entropy change (system + surroundings).
If heat spontaneously transfers from the system to the surroundings due to a finite difference in temperature between them, the entropy of the system decreases, but the entropy of the surroundings increases by a greater amount for   total entropy change (system plus surroundings = change in entropy of the universe) of greater than zero.
Hope this helps.
A: Yes, one can observe a decrease in entropy of an isolated system.
The statistics of these observation are quantified by the fluctuation theorem.  The logic of it is based on what you (the OP) suggest: since statistical mechanics is about statistical laws, one would expect there to be fluctuations.
The first paragraph of wikipedia states this well, so I'll just copy it with some emphasis added:

While the second law of thermodynamics predicts that the entropy of an isolated system should tend to increase until it reaches equilibrium, it became apparent after the discovery of statistical mechanics that the second law is only a statistical one, suggesting that there should always be some nonzero probability that the entropy of an isolated system might spontaneously decrease; the fluctuation theorem precisely quantifies this probability.

The paper in the first note (pdf) there gives access to a PRL with an experimental observation of a decrease in entropy in an isolated system. Of course, these observations were done in very small systems; as the systems get larger, the chance of observing such fluctuations decreases.
A: Quoting from this directly related post,

According to the fluctuation theorem the second law of thermodynamics is a statistical law. Violations at the micro scale, therefore, certainly have a non-zero probability.

We can calculate the entropy using the Boltzmann formula $S=k_B\log N$, where $N$ is the number of states. From the initial state $i$ to the the broken final state $f$, the change in entropy is $\Delta S=k_B\log N_f/N_i$. Therefore, $N_f=N_ie^{\Delta S/k_B}$ so that the probability of the transition from the state $i$ to the state $f$ is calculated as $p_{i\to f}=N_f/(N_f+N_i)\approx 1$, because there are several more possible micro-states in the final configuration $p_{f\to i}=1-p_{i\to f}\approx\exp(-\Delta S/k_B)$. Although the increase of entropy is a high number, the Boltzmann constant $0<k_B$ adds $10^{23}$ to the exponent, so that the value of the probability is extremely low.
