SQ will check the bags right through, I've even done it with overnight stops in SIN.

I've done SIN transit stops many times, and immigration has never had a problem with it at all. On the arrival card, where it asks for an address in Singapore, just write TRANSIT.

Actually, I'll be doing a 6:00-23:55 transit stop there this Sunday.

I always take the metro or bus into town (depending where I'm going). Its cheap.