Recently I had my second child born here in Oman, and I thought I'd record the process for getting the Omani birth certificate here so anyone googling for it might find it.
Essentially, you need the following documentation in order to get your child's birth certificate:
1. Your passport & resident card (and copies)
2. Your partners passport & resident card (and copies)
3. The letter from the hospital notifying for the birth (keep a copy for yourself as they will keep this)
4. Your marriage certificate, attested (and a copy) (it HAS to be attested by Oman ministry of foreign affairs by the way)
5. A credit card.
Now, I can only record my experience of what to do here in Muscat, I've no idea what you'd do if elsewhere in Oman, so sorry about that! Here in Muscat, once you've got all those documents together head to the Seeb ROP compound (Directorate General of Civil Status) where you got your resident card from. Go into the main building (same place as for the resident cards) and go to the reception and say you want to get a birth certificate.
The officer will check your documents, stamp that he's seen the original documents on the copies and will give you a ticket with a waiting number on it, once you've got your ticket you will go to the RIGHT hand side waiting hall. The birth certificates are handled by different staff than the resident card people, so even if the place is heaving with expats waiting for their resident cards, you won't wait more than half an hour or so.
Once your number is called, head into the office and hand over the documents to the officer. They will print a draft for the birth certificate out which you need to check very carefully - for example, my name is Peter Piper Pan, and they default to the given name of your child, and then "Piper Pan" from my name - which at least for me and my culture, is not how we do names - so check it. Once you're ok with it, they will print two certificates (one English one Arabic), apply a protective film to them and hand them to you - job done. Takes about 5 minutes.
The charge for the birth certificate is RO 3. You used to be able to pay extra and get extra copies of the birth certificates (my first born I have 3 Arabic and 3 English copies for example) but they've changed the rules now and you only get 1 English and 1 Arabic version.