JHattery

Unless you are in the very back of the bus in coach and the last to get off a large capacity plane, there's no reason you can clear immigration, retrieve checked bags, clear customs (which is generally just a walk through a door anyway), and be in queue at the taxi stand by 2am, if not a bit earlier. Taxi drivers know the landing schedules and, if possible for their schedule, line up at the airport to get desirable long distance (for Singapore) fares. Plus, they get late night surcharges which make runs from the airport even more attractive.

Being in Biz or first class, and getting off the plane quickly, proceeding DIRECTLY to immigration with NO stops (use the bathroom on the plane before descent begins, or go AFTER you clear immigration) helps. Being a local or employment pass holder helps, but even without that, so long as you have your forms properly filled out before you stand before the immigration agent, I often go from wheels on the tarmac to standing at my hotel registration desk within 45-60 minutes.

I DO NOT recommend staying at the airport if you arrive at 1AM. Secure a late arrival reservation and go directly to your hotel to avoid having to move. That way, you can settle in, shower, try to sleep, and get into a routine.