Morton's fork is used to refer to the general style of argument where any response will be used as evidence for the same conclusion. It is a logical fallacy, although it is important to note that it is not generally a logical accident but rather a devious rhetorical trick.
Morton's fork is most often noted in reference to conspiracy theories, where positive evidence is strong evidence for the theory, and lack of evidence is strong evidence that the theory is important enough that the government is eager to cover it up. However, this is also in evidence in religion (all that bad stuff is needed so that we will know evil and can thus overcome it), atheism (your direct personal experience of God is clearly a sign of mental illness -- and that mental illness also explains why you were attracted to religion in the first place), and in any political party line ever.
In many cases this is actually a case of an 'honest' cognitive basis, the confirmation bias, in which information is collected, mentally stored, and recalled in such a way that it remains entirely consistent with previously held beliefs. The difference between Morton's fork and a strong confirmation bias is entirely in how aware the person is that they are twisting the information around.
In common parlance, Morton's fork is often confused with a Catch 22; this error is understandable, as both are examples of a no-win situation. Morton's fork is an elaborate form of Hobson's choice.