The easiest part of your question is to contrast the first two with the third. You would generally use "anymore" or "any longer" after using a negative form of a verb, whereas you would use "no longer" with a positive form of a verb. Here are examples: I do not believe that politician anymore. I do not believe that politician any longer. I believe that politician no more. The first two versions would be more common than the third, and "no more" is a stronger rejection of what used to be. As to the difference between anymore and any longer, you will hear anymore in the US and Canada more, but any longer would be more common in Great Britain. They have the same meaning, but "any longer" is a bit more formal.