Dear Koala-Mummy
Just to share, my colleague pays $700 per month for her babysitter. She sends the baby to the nanny's house before 8am and pickup before 6:30pm. Besides taking care of the baby, the nanny will also do the baby's laundry, clean the bottles, etc - there is a set of baby stuff at the nanny's house. She does not bring the baby over on weekends and on her off-days. So the nanny works 5-day week. When the nanny is sick, or when the nanny has to run errands or go for holiday, they just make arrangements to let the baby stay home with them.
One of the reason most nannies like to work from their home is because they get to watch their other children, or they get to do all their usual housework, incl cooking and cleaning for their own family. I'm afraid $500 is too low for you to get anyone good or committed.
That said, you can try to put your baby at infantcare such as Sparkletots. For SG citizens, there are special rates. After the normal subsidy, you usually pay around $700+. If you qualify for additional subsidy, then it could be as low as $10/mth. It is a good deal, it is safe (teachers are pre-qualified and pre-checked by the centres), and the timing suits you cos they open from 7am to 7pm.
All the best.