A funeral home would be able to assist with this as they would lodge an application at the nearest National Registration Department on your behalf. However, if you are in a rush, it's advisable to apply yourself as the next of kin. The department's more likely to issue the certificate earlier when the applicant is the deceased's next of kin—or that was what I heard.
So, if you intend to apply personally, visit the NRD's website for the specific checklist before you head over with the documents. You could save the extra time of going back and forth repeatedly.
For me, since I was paying the funeral home, I figured it would be better to let them handle everything as they are the experienced folks. It took quite a while, but it didn't matter as I was able to wait. (I honestly didn't want to run around, nor did I have the patience to do so.)
That said, before you proceed to get the death certificate, ensure that you have copies of the deceased's IC. The NRD would destroy the original IC once the death certificate is issued (to prevent fraud, I'm sure.).
And yes, it's always better to have extra copies lying around.
Once you have the original death certificate, the first port of call is to have copies of this document before storing it in a safe place. You'd need it virtually for everything bearing the deceased's name—cancellation of accounts, cancellation of prepayments, and the grant of probate/letter of administration for properties, among others.
(As for the grant of probate/letter of administration, you're better off dealing with the lawyer on this. Even better, do it when the deceased is still alive and has legal capacity to make uninfluenced decisions.)
Any supplementary documents bearing the deceased's name shouldn't be destroyed as long as they used to own properties while alive. It'd be safer to prove their identity and ownership that way, too.
Final Thoughts
It'd be much better to deal with matters of death while the deceased is still alive. Trust me, a lot of unnecessary headaches can be avoided.