You can doze off at the T3 lounge (as mentioned, they would give you a blanket and pillow), but be forewarned that it's very busy until 2am. After that, the lights do not go off, there's light music in the background, and there's some cleaners who would be going around. Just fine if you only need a couple of hours rest and want to save $$$, but not ideal if you need to be well-rested for the next day. Bring an eye mask.

As for safety outside the airport - SG is extremely safe. Too safe, in fact