Actually they aren't prohibited in Islam. Read this article: http://www.quran-islam.org/articles/dogs_%28P1159%29.html
It is traditional among Muslims all over the world to regard the dog as a dirty animal that when touched would void the wudu (ablution) and infect the one who touched it with "nagasah" (dirty impurity)!
Sadly, this concept comes from fabricated hadith which claims that the Prophet ordered the killing of dogs and gave numerous hadith that prohibit the keeping of dogs except for hunting and guarding, due to their dirty status!
However, by studying the Quran we find no such truth. No where in the Quran are dogs prohibited, nor is there any mention of any contaminating effect of these lovely animals who are man's best friend. Consequently, we must dismiss all these hadith that fabricate lies against the Prophet.
1- God tells us in the Quran about the story of the dwellers of the Cave (Surah 18). In verse 13 God tells us that they were good believers and that God guided them. In verse 18 God tells us that they had their dog with them.
Now if dogs are prohibited and dirty, would God speak of those dwellers of the Cave (who had a dog) as good believers?
2- In 5:4 God tells us that it is okay to eat what the trained dogs catch (dogs are used in hunting). Now if the dog is an animal which causes contamination by mere touch, would God tell us it is perfectly okay to eat what the dog catches with his mouth (let alone just touch the dog)?
3- The Quran contains a very important rule for all believers, and the rule is:
Nothing is haram (unlawful) unless it is prohibited by God Himself, and since God describes the Quran as complete, perfect and fully detailed, thus all the prohibitions decreed by God are found in the Quran. The following Quranic verses confirm this truth:
“Say, "Who prohibited the nice things God has created for His creatures, and the good provisions?" Say, "Such provisions are to be enjoyed in this life by those who believe. Moreover, the good provisions will be exclusively theirs on the Day of Resurrection." We thus explain the revelations for people who know.” 7:32
“Shall I seek other than God as a source of law, when He has revealed to you this book fully detailed?” 6:114
“You shall not utter lies with your own tongues stating: "This is halal (lawful), and this is haram (unlawful)," to fabricate lies and attribute them to God. Surely, those who fabricate lies and attribute them to God will never succeed.” 16:116
There is not one verse in the Quran where God says that dogs are dirty or that they are prohibited to keep.
4- Is it logical that God would create an animal to be man's best friend and serve him in many ways, and then prohibit such an animal?
5- If this is what the Quran has to tell us about the issue of dogs, then where does all the prohibition come from? Where do the claims that dogs are dirty animals which if touched would void our ablution and render us impure come from? As most other corruptions which have crept into Islam, the source is always the fabricated hadith which are falsely attributed to prophet Muhammad!
Hadith tell us that the Prophet prohibited the keeping of dogs as pets. Other hadith tell us that angels won't enter a room where there is a dog. Other hadith tell us that if we touch a dog our ablution is void and we become impure, and that we have to wash seven times to clean this impurity, the final time in dust. That seems inconsistent with the Quran saying you may eat what they catch for you! Some other hadith go even beyond that to say that we must kill all black dogs!
The following are some examples:
Malik's Muwatta, Book 54, Number 54.5.13:
"Malik related from Nafi from Abdullah Ibn Umar that the Messenger of God ordered all dogs (other than sheepdogs or hunting dogs) to be killed".
Ibn Hanbal's collection: The Messenger of God said:
"You shall kill all black dogs, because they are devils."
The question is: Did the Prophet really issue these prohibitions? In addition, did the Prophet have the authority to issue these prohibitions?
The answer is given in the Quran:
"O you prophet, why do you prohibit what God has made lawful in order to please your wives? God is Forgiver Merciful" 66:1
Obviously God did not include the words in 66:1 in the Quran to belittle the Prophet in our eyes, but these words are placed in the Quran in order to confirm to all believers across all time that the Messenger of God does not have the authority to prohibit anything which is not prohibited by God. God is the only Law Maker (6:114) and the only duty of the messenger is to deliver God's message (5:92).
The outcome of all the above is that we must discard all the lies attributed to the prophet regarding prohibiting dogs and be focused on the Quran as the only source of guidance and religious law.