Hi betonic,
There's two approaches. One is to stay at the airport, the other is to go into town.
As NickC suggests, the most obvious answer are the transit hotels. run by Ambassador. There is one in each of the three terminals, all on the airside.
Check this link:
http://athmg.com/
Now you haven't given us details of your itinerary, especially the carriers involved (ie, which airline is bringing you to SIN, and which is taking you to Bali). If you have status with any of them, you might be able to enter one of the airline lounges while you wait for your connecting flight. Also, there is the Rainforest Lounge at Terminal 1. This is a pay lounge where it doesn't matter who you're flying with. However if these are not options, note that Changi Airport has no shortage of things to do. Unlike most airports, Changi is almost like Disneyland in the attractions available (nb - airside, not groundside). To be fair I can't assume that everything is open at 3AM but there should be enough things happening to keep you distracted. Also there are sleeping areas with lounging chairs (with footrests) so you can actually catch a few zzzz's. Also note that if you are flying into or out of the Budget Terminal some of my replies may have to be adjusted somewhat.
Check this link for Changi Airport:
http://www.changiairport.com/
Link for Changi Airport lounges (nb all airside):
http://www.changiairport.com/in-transit/airline-lounges
As for payam81's suggestion of going to Mustafa Centre (you'll have to go through immigration and customs and legally enter Singapore, then take a cab) you can try this if you're adventurous (yes, you have enough time - you can do a cab there and a cab back, it's in Little India) but I wouldn't. Knowing that there's another two hour flight to go, I'd stay put. But you should decide for yourself. Here the link for Mustafa Centre:
http://www.mustafa.com.sg/index.asp
In any case, I feel that Singapore is best seen during the day (the casinos excepted). The middle of the night, most people are indeed sleeping.