If the definition of socialism is free market regulation and the capitalist index is measured by how much the government regulates the market then how the hell is regulation a part of capitalism? Again, this is a mixed economy you're describing.
Your idea of self regulating works ONLY if humans are not involved. It's the same thing with communism as with capitalism, the extremes NEVER work.
The definition of Socialism is not free market regulation. The definition more accurately would be community owned and controlled. Thus, 3rd parties can regulate and control transactions that they are not directly involved in. In Capitalism, the regulation would have to be between the two transacting parties. You would still have a 3rd party to enforce contractual agreements which will be the state. But it would be incorrect to say that Socialism is freemarket regulation. By definition its not free market.
All examples that exist are based in realities that exist, yes of fucking course. In a reality where there was a nation without any regulation on a free market there would be examples of capitalist created monopolies but since that isn't actual reality I can't give real world examples. There have been PLENTY of examples of attempted monopolization of the market though, usually via price dumping to get rid of the competition (MS vs. Netscape is an example) when they didn't want to sell their entire idea for cheap.
Now regulatory powers used by corrupt people is a problem (as is corrupt coppers) but doing away with the entire thing doesn't solve the problem, getting rid of the corruption DOES.
MS vs Netscape is a great example to prove my point. Yes, MS tried very very hard to kill competition. They tried to control and manipulate transactions they were not directly part of. This is an attempt to do monopolistic activity. That said, where is MS now in the browser war? The reality is that MS did not lose the browser war because the government stepped in to fix it. They lost due to competition. The main reason MS was able to beat netscape was because they went free which is actually a competition move.
And, again, you can isolate the factors that we are talking about here. Its true that we do not have any pure form of capitalism or socialism, but that does not matter. You can look at individual monopolies and look at what is sustaining them and see that its not because of lack of regulation.