Registering a birth in Singapore

The hospital will assist in the birth registration. Just submit the following documents to the front desk: IC cards (for foreigners: Passports with visas) and birth certificates of both parents, marriage certificate if applicable. There's also a form to be filled, which the hospital staff will give you.
You will receive your child's birth certificate soon after submitting. Depending on your home country's regulations, you then need to translate, legalise or register it with them.

@Beppi - Thank you for the sharing of information

Hospital assists only during the first few days (7 days as far as I remember), afterwards, you have to go to ICA (in Lavender) yourself.
Both parents must be present, or you have to provide a translated marriage certificate.
Prepare both parent's passports, Entry Permit, and Disembarkation/Embarkation cards.
Each baby born to foreigner parents will be issued a special pass for 42 days, during which, either a valid passport & visa should be issued, or at least a proof that his parents are making his formalities at his embassy, in order to extend the special pass.

Ah, one more advice out of my experience, make sure you have the right spelling of your child name and that you revised EVERY SINGLE WORD in the birth certificate well before signing it.

Refer to this page for details
http://ica.gov.sg/page.aspx?pageid=186

Please also note that being born in Singapore does NOT give the child any right to citizenship or visa (except the 42 days special pass). If it cannot get a visa through other means (e.g. as dependent of parents with long-term residence visa in Singapore), it will have to leave the country.