I've stayed air side overnight after arriving at Changi many times, and connected air side or headed out through customs/immigration the next day, with or without checked baggage. It's a smooth process.

Transit hotel economy room is pretty good value, as it includes shower.

Otherwise there are free nap chairs, or the free movie theatres !

http://www.sleepinginairports.net/asia/singapore.htm

Air side transit desk for UA doesn't open until a couple hours before flight departure.