The only one in the Gulf region so far (I think) is the Sharjah based Air Arabia, and they seem to have already made a significant impact on the market. I went to the websites of Air Arabia, Gulf Air and Emirates today to compare prices. This is what I found out. To fly from Bahrain to Dubai costs BD35 (not including tax) on both Gulf Air and Emirates. Air Arabia does not fly to Dubai airport, so they fly you to Sharjah, and then put you on a coach to Dubai airport (which takes about 45 minutes). This route will take a bit more time, and is somewhat inconvenient, but it costs only BD12 ! (BD11 for the flight to Sharjah + BD1 for the coach to Dubai). Sounds like a good deal to me. I think that even the bus services from Bahrain to Dubai would find it hard to compete with that price.
Good luck to Air Arabia and all of the potential new low cost carriers.