I disagree with the previous answer. (And I believe it does not answer the question since the question asks why Singapore is a dictatorship while the first answer tries to talk about if it is a dictatorship...) I would argue that Singapore has essentially been a dictatorship (or at least a very strongly authoritarian country with one man holding a dominant position) under Lee Kuan Yew.
To quote Fareed Zakaria from the foreignaffairs.com link below
Singapore’s government can best be described as a "soft" authoritarian regime, and at times it has not been so soft. He was prime minister of Singapore from its independence in 1959 (it became part of a federation with Malaysia in 1963 but was expelled in 1965) until 1990, when he allowed his deputy to succeed him. He is now "Senior Minister" and still commands enormous influence and power in the country.
In my opinion, the reason that Singapore got to be this way is a combination of the personal characteristics of Lee and the fact that East Asian countries seem to have an affinity for allowing autocracies. (Please note that I am of Filipino ancestry and would assert that the Philippines shares this affinity. So I do not think I can be accused of anti-Asian racism...) There is clearly something in East Asian political culture that has allowed dictatorships and single party governments to arise in many countries in the region.
That said, I would argue that Lee's personal abilities allowed Singapore to be the stable and prosperous country that it has become.
So I would say that East Asia's political culture helps make a dictatorship possible while Lee's personal qualities made it a very successful dictatorship that has done pretty well for its subjects.
We have to start this answer with the correct facts. Singapore is not a dictatorship; it is a fully functional parliamentary democracy. It has elections once in four years and has compulsory voting, so all those who are eligible for voting have to cast their votes.
The party that has been winning elections since Singapore turned to Democracy is the People’s Action Party or PAP, but there are other parties that play the role of opposition parties. These include the Singapore Democratic Alliance, Workers’ Party, Democratic Progressive Party, etc. Though the fairness of elections in Singapore has often been questioned with the PAP accused of taking harsh action against opposition parties to discourage and impede their success, which include gerrymandering and filing of civil suits against them, Singapore on a whole has been quite successful in maintaining its political structure.
The excellent planning has allowed Singapore to become the 5th richest country in the World, on a per capita basis and among ten least corrupt nations. The nation is developing very fast and is able to attract foreign businesses and investment on a large scale.