Their government picked some key industries like semiconductors and invested a lot of resources towards training a workforce and stimulating business. It has paid off in huge dividends. All done democratically.
Additionally their recent environmental push is driven by local people and very democratic - individual recycling rates have skyrocketed. It will be interesting to see if they can do the same with renewable energy.
Similarly for Singapore. I understand China to be a lot more capitalist than India for example. It is true that Japan and a lot of european countries have done better with democracy, however they typically have a lot homogenous populations - very unlike of bigger countries like the US, China and India and also unlike some smaller countries like Singapore.
Even then there is little reason to believe if a communist form of government would not have done better in these countries also.
It is a very Confucian form of democratic society, but democracy it is.
I do think that China's more enlightened leaders are probably hoping that they'll be able to use Singapore as a model for their own society at some point in the future.
India is a democracy, but the state is hardly transparent, tolerance has given way to appeasement for vote politics and the court/legal system does not work anymore.
Moreover, China's growth is nothing compared to Japan's economic miracle. The fact of the matter is that China has far better national resources than Japan ever did, and should have succeeded decades ago. It was the early leadership that held it back.