Tattoos are hip. No doubt. There are certain places in the world that are renowned for their tattoos, but Dubai really is not one of them. As far as I know, I think tattoos are illegal in Dubai. But this certainly does not prevent people from getting tattoos in Dubai.
Getting a tattoo in Dubai can be quite an ordeal. You should know the right people and you should know how to approach them. Firstly, most of the tattoo parlors in Dubai use natural components and they are washable. If you are looking for a permanent tattoo, then you should either try the parlor near Madinat Jumeirah or the Beach centre. As far as my knowledge goes, I think that the above two mentioned places offer permanent tattoos and they are pretty much the best you can get in Dubai.
Also 4 leaf clover had their outlet in Diyafah Center, but now they have moved to Bur Dubai. I am not sure about the exact location, but if you look up the local directory, I am sure that you can find their listing. They also have contacts throughout Dubai and they can get you hooked up with some really good parlors. But be warned, i think that tattoos are not legal any more in Dubai.
Probably with the opening of the Atlantis and the Restless City, I guess we can hope to get more tattoo parlors in Dubai.