The Arab World has one completely failed state: Somalia, which is not even identifiable as a state anymore (or Arab for that matter). It has one nearly failed states: Sudan which has staved off Somalization by giving the Southerners their freedom (unlike Abraham Lincoln). Then there is Yemen. As for Yemen? I believe it will become an unidentifiable state, like Somalia, possibly with the South regaining the independence it foolishly gave away in 1990 to join the tribal North. Northern Yemen is truly in danger of falling completely apart, American drones and Saudi war planes notwithstanding.
No, Libya is not likely to become a failed state. It may try to become a failed state during the next few years, many African states that were ruled by long-term despots have headed that way. But Libya is a potentially rich country with a relatively high degree of national identity. Besides, it is too close to Europe to be allowed to fail as a state: who is going to stop all them boats?