On this ICA page 103 (http://www.ica.gov.sg/page.aspx?pageid=103) it says,
(bolding mine)Pregnant women who wish to deliver their child in Singapore must apply for permission through a local sponsor before their arrival.
Following the link on that page, you reach page 237http://www.ica.gov.sg/page.aspx?pageid=237. On this page it says
(bolding mine)Female visitors may apply for a Visit Pass (Long Term) for the purpose of delivering a child in Singapore at the Visitor Services Centre, ICA.
So on one hand it says to apply before arrival, then it says to apply at the ICA in Singapore.
I know that the rules changed about a year ago where previously all pregnant women over 6 months needed permission to enter even for a visit and now they don't, but I'm not certain that women intending to give birth in Singapore can just turn up and enter without prior permission. Does anyone know?