The most famous examples of this type of growth were called the Asian Tigers. As you noted Singapore was one of them. There were another three , these were South Korea,Taiwan and Hong Kong. All of these countries entered the post war era in a very poor state. All were very poor countries however they basically reformed their economies in the 60s and 70s having growth figures of over 7% which is very high. They heavily invested in Education which allowed for them to produce more and more complicated goods.

That's actually quite a hard question to answer, as the criteria that delineate a 'developing' country from a 'developed' country are actually quite arbitrary. Would you consider Cuba a developed country, given that, according to the Human Development Index it has 'very high human development'?

As has already been mentioned, the Asian tigers are probably the most famous example of countries that have undergone rapid development since WW2, as well as the Gulf states (particularly the UAE, Qatar and Kuwait) which used their vast oil reserves to go from being economic backwaters to being some of the richest per-capita nations in the world. Among European nations, Spain has already been mentioned, but Ireland is also a good example of a poor European country which 'got rich' during the 90s, and some people consider Poland and the Baltic States to be currently undergoing the same process despite a painful economic contraction during the late 00s. If you define Europe according to its Eurovision criteria, you might also include Israel as a country that has entered the developed world since the War.

You also have other countries like Malaysia and Mauritius which are much richer than they were a generation ago, but still maintain a level of development significantly lower than most Western countries. And then you have the nations which went from rags to riches in the 80s and then very tragically went back to rags again, such as North Korea or Nauru

tl;dr it's complicated.