Private residential apartments cannot be rented out for less than 3 months.
Public housing cannot be rented out for under 6 months and each application requires approval from HDB.
Your budget is way too low for a month even for hostels.Edited: 18 December 2017, 11:18
Try the hostel with toilet ensuite
Increase your budget to $350+ and you might be able to get a hostel stay for a month. At $250, I really don't think there are any hostels offering that sort of rates for a month long stay.
Legal stay with SGD250 per month as a tourist is almost impossible. Those with work visa, companies do find them about SGD180 per month but is 4 to a room.
<<250SGD for 1 month.>>
You have to double that budget to stay in a hostel for a month. All airbnb rentals are illegal in Singapore for less than 3 months lease,please beware of that and steer clear from those listings.