The second law of thermodynamics states that **global entropy of an isolated system always increases or remains constant**

This tells us a couple of things that are relevant here:

 - If the system is not interacting with anything, entropy cannot decrease. I.e. the entropy of the universe will never decrease without violating the laws of physics as they are currently known
 - If you have a sub-system of the above isolated system, the **entropy of the sub-system can decrease. However, this causes entropy somewhere else to increase so that the total change is either positive or $0$**

A simple example of a decrease in entropy of a sub-system is taking some water vapour and cooling it down to become liquid - you've taken heat out of a system. This heat has gone somewhere else and increased entropy there, but the entropy of the water vapour has decreased. A fridge is a good example of a physical object that can cause this to happen.

A white hole is different, as it (I think) decreases the entropy of the universe, so invalidates the second law of thermodynamics and so are assumed to not exist because:
>    “The law that entropy always increases, holds, I think, the supreme position among the laws of Nature. If someone points out to you that your pet theory of the universe is in disagreement with Maxwell's equations—then so much the worse for Maxwell's equations. If it is found to be contradicted by observation—well, these experimentalists do bungle things sometimes. But if your theory is found to be against the second law of thermodynamics, I can give you no hope; there is nothing for it but to collapse in deepest humiliation.” Sir Arthur Eddington (The Nature of the Physical World, 1915)

In other words, anything that increases order in any way gives a decrease of entropy of whatever is becoming increasingly ordered, but at the cost of increasing the entropy somewhere else.

So to answer all your other questions: We already live a world where entropy of small regions can decrease!

Interesting aside: It is possible for a "broken vase to spontaneously un-shatter itself", it's just that the probability of such an event occurring is so small that you would have to wait longer than the age of the universe for it to have any reasonable probability of occurring

Edit: I've re-read the question and feel that I should add the below:
Things like kicking a pile of gravel and the result being smaller piles of pebbles is 'impossible' due to probability, not entropy. That is, as per 'interesting aside', it's perfectly possible for such a thing to happen, it's just that the probability of it happening is tiny when compared with the probability of it not happening. Saying that an isolated system tends towards maximum entropy is essentially the same as saying that the system tends towards the most likely outcome, because they are the same thing. The above quote from Eddington is true, not because of physics, but because a system increasing in entropy is a system tending towards the most likely outcomes, which is unavoidable by definition. So, if you want to decrease the entropy of something, you can do it easily (e.g. pour boiling water into a mug, then put it in the fridge). If you want an area where this happens by itself, then you need to either travel backwards in time, a white hole or something similar, which is all assumed to be impossible (to current knowledge) because they violate the second law of thermodynamics. Quantum physics does however do weird things with probabilities (see e.g. the [Wigner function](https://en.wikipedia.org/wiki/Wigner_quasiprobability_distribution)), so who knows what's actually possible on the tiniest of scales? Using quantum physics to decrease entropy of a subsystem is entirely possible - http://www.nature.com/nature/journal/v474/n7349/full/nature10123.html - but again, the 2nd law of thermodynamics isn't broken.

To sum up, entropy is probabilistic by definition, so things like vases fixing themselves doesn't happen, not because of physics, but because of probability. However, decreasing entropy of a subsystem (at the expense of increasing entropy somewhere else) is an everyday occurrence.