Dry Season: May - July
Rainy Season: November - January
Singapore's weather is relatively the same throughout the year.  It is hot and humid with steady rainfall.  November to January mark the wettest months while May to July mark the driest.
However, realistically to the foreign tourist, there is little distinction between the seasons, so in regards to the weather, any time is a good time to travel to Singapore.
Singapore consists of a variety of ethnic communities so visitors should consider the festivals and events that are happening at different times of the year. The main cultural festivals are: Chinese New Year, Hari Raya, Deepavali, Thaipusam, and Mooncake Festival. The respective colonial ethnic quarters are decorated, and temporary market stalls and bazaars are erected among the festive events.
It should be noted that the majority of these celebrations depend on Lunar Calendar dates.
If you are an avid shopper consider the Great Singapore Sale held June to July. Formula 1 racing takes place in September and it is a major event attracting fans from all over the world.
Chinese New Year takes place sometime between January and February, depending on the moon's cycle.  However, because of the popularity of this celebration, it is considered "peak season" and not the best time to negotiate favorable travel rates.