One of the nicest things to see when in Dubai...is not actually in Dubai.
About an hour's drive from Dubai, in the city of Abu Dhabi, one can find the Sheikh Zayed Grand Mosque. It's not only the largest mosque in the Middle East, but also perhaps the most beautiful. The mosque is especially stunning when light shows are on. The Grand Mosque is one of the few mosques in the UAE which are open to non-Muslims.
All women visiting the mosque should wear loose fitting clothes and cover their hair, but black overcoats (abaya's) and headscarves (sheyla's) are available upon request.
Please check the mosque's website for more info on opening timings, (free) guided tours and more.