Hi poodlek
My advice is to google, google, google! And there is info on forums on this website too. My husband and I have very basic coverage
through his employer but I was also directed to Pacific Prime who I think are based in HK but have an office here.
In regard to public and private hospitals, I'm not 100% sure of the difference, apart from price. Because I'm choosing to have a water birth, I'm looking at NUH (public) and Mount Alvernia (private) as they're the only two that have facilities. Haven't had a tour of them yet but of course public will always be cheaper. Although NUH is cheaper, I have heard it's still very good but will let you know after I look at both!
I'll be having a doula (you have to for a water birth but I'd like one anyway) and from what I've been told (by a midwife), midwives are unable to deliver in Singapore (unlike Australia) so you'll always have to have an ob/gyn do the delivery. There are quite a few doulas around but once again, the internet is your best friend.
Hope this helps a tiny bit. And I hope it's all correct! It just what I've researched.