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 occuring is so small that you would have to wait longer than the age of the universe for it to have any reasonable probability of occuring