Although Arabic is the official language of the UAE, English is the most spoken language in Dubai. You will also find a lot of expatriates who speak Hindi, Urdu, Bengali, Tamil, Tagalog, Persian, Chinese, Malayalam, in addition to many other languages.
Arabic is the official language for both written and verbal communication in Dubai. Government offices and affairs are conducted in Arabic, and rules and laws are published in Arabic.
Arabic is mainly spoken by the natives of the UAE, called Emiratis. You can get by with just speaking English in Dubai, but if you are ready to learn Arabic you will probably be more welcomed by the Emiratis.
English is the most widely spoken language in Dubai, especially for business, trade and tourism. Around 75% of the population in Dubai are expatriates, most of whom speak English and/or their native language. There are a lot of different nationals working in Dubai, so you are likely to find someone who speaks your language.
If you can speak a South Asian language (e.g. Hindi, Urdu, Malayalam, Bengali and Tamil), you will have the opportunity to practice in Dubai. Many expatriates are from South Asia and some of them only speak their native language.
The problem of the growing number of English speakers has been raised during discussions for the Dubai Strategic Plan for 2015. It is a project for Dubai's development proposed by the UAE Prime Minister and Vice President. The government is afraid that the UAE is losing its national identity.
Dubai is a really cosmopolitan city with a lot of different cultures. As it attracts more and more tourists each year, speaking multiple languages becomes more and more important for Dubai’s residents.
You will find most of the road signs, boards, etc. in both Arabic and English. However, if you plan to leave Dubai to visit the rest of the UAE, keep in mind that in other cities and towns English is not so common. Most rural people don't understand English.
Public schools usually teach in Arabic, with English as a second language. Most of the private schools have lectures in English or in another foreign language.