Hmm 3K is still decent, depending on your expectation on living conditions.
And no, housing allowance is less likely for your range. So you gotta find your own place.
If you rent a single room in a HDB, it will be between SGD400 to 1000 depending on where you live. The closer you are to the city or the popular spots (like Holland Village), the more expensive it will be.
Double that amount for a single room in a
condominium.
So where you will be working will play a role in this.