In a poor economy it would be inadvisable to quit a job before you have a new one. In a US state where you work "at will" they can fire you for almost any reason. You take the risk that they will let you go the moment you tell them you are looking.
This doesn't happen very often, but what does happen is that they can start to think of you as being disloyal or only going to be around for a short period of time. They won't send you to training, or add you to the new project.
Telling them you are looking does mean that in some cases they will offer you a raise, a promotion, or a transfer. Some times that is just a delaying tactic, and then they still view you as disloyal. In the end you may delay your departure for a while but ultimately you end up leaving.
You have no requirement in the United States to tell them, and little practical reason to do so.
Most companies you interview with will avoid getting a reference from your current company, especially if you let them know your current boss is unaware you are looking.