On the airlines website, knives are listed as restricted items (but are not prohibited). I do not have any checked in luggage and am travelling from Darwin (Australia) to Singapore in Singapore Airlines. Can I carry a ceramic knife with me ?
No. Restricted items may only be carried in checked luggage. However, Singapore Airlines permits you to check at least 30 kg for free, so I would suggest you simply check your carry-on luggage.
Knives don't go in carry-on, period. (maybe really small ones, depending on the country.) The material the knife is made of doesn't matter. Ceramic, wood (ebony will hold a nice edge), bone, flint and so on have all made fine weapons millenia before steel.
If you do pack a non-metallic weapon in the darker corners of your carry-on, and they find it, you could be up on an "artful concealment" charge, which is usually very serious. In contrast, a Crocodile Dundee grade Bowie knife will usually not cause any substantial grief - hiding the beast is impossible, you'll just get pointed back to baggage check.
Anything that looks like a weapon is not allowed in the carry on, and this is pre 9-11. A long time ago I once caused quite a stir at LHR and EWR with a key chain shaped as a gun, the size of a thumb.