Is there a correlation between the colonial power and the stability/success of the post colonial state?

The Portuguese colonies were amongst the most developed. It was the scramble for their mineral resources, namely Angola's oil, which led the superpowers of the time to destabilize them, leading to civil wars and economic decline.

As someone who studied in Indonesia (former colony of Dutch) and Singapore (former colony of British), I can say very well the difference lies in the education investment made by the British.

This simple slides (slide 21) shows very clearly the investment made by the British in their crown colony:

The Dutch, however, did not do the same for its colonies (i.e. Indonesia), since the main target was to utilize the colony to increase revenue of the VOC. There are numerous article on this issue if you can read the local language. Technically, even the textbook recorded the absence of education service for the native, unless they were willing to 'cooperate' with the Dutch.

Post-colonization-wise, it depends on the nation itself, whether they are able to utilize the 'advantage' of their education. There are cases where the educated generation is rendered meaningless due to (civil) war, too.

Why ignore Australia, New Zealand & the United States - are they are not all former colonies of Great Britain?

The United States does particularly well, being the country with the highest GDP. They gained their independence in 1789, that is two centuries ago.

One may suppose, that most of the former colonial nations that aren't doing so well are still building their political institutions. This is the work of several generations. After large-scale economics requires strong political institutions.

One could also perhaps point out that these countries could industrialise relatively rapidly as this is the social environment that colonists came from, and who form the dominant & majority population. Whereas places like Africa or India this 'lesson' in modernity would have had to be absorbed by the indigenous population.

I conducted a small survey of the (quite extensive) literature on this topic , and found that there are many different conclusions as to both whether and why the identity of a colonial power had an impact. Quite frankly, the number of contradictory papers I found indicates that there is no scholarly consensus on this issue, and that it remains an ongoing debate (most of the papers I found were trying to rebut opposing claims, and found totally different things with the same data). If you're curious, here is a small sample of some of the more interesting papers I found:

[Please note that I have academic access to scholarly databases; most of the links here are behind a paywall]

Blanton, Mason, and Athow (2001), in their paper Colonial Style and Post-Colonial Ethnic Conflict in Africa compare British direct rule with French indirect rule and conclude:

Results indicate that, after controlling for other salient factors, a British colonial legacy is positively associated with ethnic conflict.

Bernhard, Reenock, and Nordstrom (2004) in The Legacy of Western Overseas Colonialism on Democratic Survival conclude that:

We further find that the legacy of specific colonial powers has an important effect on survival as well. Unlike previous studies, we find that former Spanish colonies outperform British colonies when colonialism is conceptualized holistically. However, when we break colonial legacy into separate components (development, social fragmentation, and the relationship between the state and civil society), we find that the advantages former British colonies enjoy are attributable to the legacy of the state/civil society relationship. Moreover, we show that at least in the case of former British colonies, time spent under colonial rule is positively associated with democratic survival.

Lange, Mahoney, and vom Hau (2006) in Colonialism and Development: A Comparative Analysis of Spanish and British Colonies:

Our argument specifically shows that the historical processes through which colonial institutions were installed and shaped subsequent development differed dramatically for Spanish and British colonialism (see fig. 2). Spain colonized most heavily precolonial regions that were prosperous because these areas offered the greatest potential for accumulation under a mercantilist economic model. By contrast, Britain colonized most heavily precolonial regions that were less complex because these areas offered the greatest potential for capitalist accumulation. In turn, areas that were heavily colonized by Spain saw the introduction of substantial mercantilist institutions, and these institutions became important impediments to postcolonial development. Areas that were heavily colonized by Britain saw the introduction of substantial liberal institutions, and these institutions were positively associated with development. Hence, both Spanish and British colonialism reversed the fortunes of precolonial regions, but they did so in very different ways.

In Colonial Legacies and Economic Growth, Grier (1999) finds:

I also find that the level of education at the time of independence can help to explain much of the development gap between the former British and French colonies in Africa. Even correcting for the length of colonization, which has a positive influence on education levels and subsequent growth, I find support for a separate British effect on education. That is, the data imply that the British were more successful in educating their dependents than were the French. The potential for expanding this work is enormous. Broadening the sample to include the 63 countries from the original sample might help to explain differences in Spanish and British post-colonial development. Looking further at initial and current education rates could help illuminate the impact of education and its persistence over time.

This poses an extraordinarily simplistic question. The histories of different 'colonies' are so utterly varied in their type and circumstances that it would be almost impossible to find useful examples for a contrasting case study. And what would be the point anyway?

'Colonies' which did particularly well, both before and after independence, are ones where there were large settler populations. For some reason 'settler communities' tend to make strong economic progress. The United States is such a 'settler community' almost entirely made up of migrants since the 18th century. Canada, Australia and New Zealand are 'settler countries'. But so is Singapore and a large portion of the population of such places as Malaysia, South Africa etc.

The settlers in Singapore and Malaysia were Chinese and Indians, who came to make their fortunes. Today they form the larger share of the bourgeoisie of those countries. The indigenous Malay population has been slower to urbanise and to industrialise.

What Britain and her liberal Empire mostly (with some obvious exceptions) provided was a good system of law and commerce and a banking sector in which business could thrive, and in which people could retain the profits from their work and endeavour. They also provided a police service and mostly a peaceful state.

Problems that have arisen on decolonisation have mostly been in those large numbers of cases where there were divided communities. As the colonists left there was a often a struggle for control between different factions. Hence since 1945 British forces have had to stand between Malays and Chinese in Malaya, Greeks and Turks in Cyprus, Jews and Moslems in Palestine, Hindus and Moslems in India, Africans and Asians in East Africa, the Indigenous people of South Arabia (Aden) and the settlers of the port city of Aden. There are plenty of other examples.

I am vastly less well informed on French and Spanish colonisation/decolonisation.

This has not answered the question because I believe that there isn't one. But it may point to a few matters that need to be considered.

There is in fact a small PDF available online which attempts to answer this question through detailed study of a small area of the South Pacific. The islands of Vanuatu were administered jointly by the British and French.

The author finds that "political indicators are in favor of British administration, but economic indicators are in favor of French administration".

His paper is called Is British colonization better than that of the French?: A study of Vanuatu and it's here:

http://eh.net/eha/wp-content/uploads/2013/11/Yoo.pdf