I would definateley go for Singapore, i dont think the diving even compares in Dubai. Singapore is a whole lot cheaper as far as a destination goes... shopping, hotels, food etc. It is a very pleasant destination with a multitude of things to do and see, beautiful islands to visit and the diving is unlimited!
There isnt much to do in Dubai besides shopping (very expensive, even during the so called shopping festival... Im finding even the shopping in Australia much cheaper) desert safari, gold souk (get harassed continuously to buy water etc outside shops, get stared at and followed if you are a woman), high tea at the Burj, and lunch/dinner at restaurants in hotels... and thats literally about it!
However if you come early december the weather is great and the Dubai Rugby Sevens is on and it is sooooo much fun.
The diving in dubai is pretty crap at best. Like Kadleck mentioned Fujairah (about 1.5 hrs drive from Dubai) and the Musundam in Oman is great.
The best dive center there is Freestyle Divers in Fujairah (Dibba) They have day boat trips out to Musumdam, and a reef with lots of turtles at Dibba Rock directly opposite the centre based at the Royal Beach Hotel Dibba. It is owned and run by an English man, the gear is clean and well maintained and their prices are good.
The Rotana Fujairah (dibba) or the JAL Hotel in Fujairah (dibba) is reasonably close and much much nicer to stay at then the Royal Beach hotel, about the same price, brand new and serve alcohol (bonus!).