A self-refuting argument is one which must be false if it is true.
The excellent blog Tough Questions Answered has a list of self-refuting arguments: Are You Refuting Yourself?
“No English sentence is longer than three words.”
“There are no truths.”
“I do not exist.” The problem here is that a person must exist to make the statement that they do not exist.
“Anyone who is biased should not be trusted.” Isn’t the person who is making this statement biased himself?
“Only science gives us true knowledge.” How do you know that statement is true? It isn’t a statement of science.
“All truth is relative.” Is that truth relative?
“There are no absolutes.” Is that statement absolutely true?
“It’s true for you, but not for me.” Is that statement only true for you or is it true for everyone?
“You should be skeptical of everything.” Should we be skeptical of that statement?
“You can’t know anything about God.” Do you know that God is unknowable?
“You ought not judge.” Isn’t that a judgment?
“You should be tolerant.” Aren’t you being intolerant of me?
Some of these are commonly used by atheists/agnostics/skeptics (e.g. “Only science gives us true knowledge”) but I see them more commonly coming out of postmodernism. Of course, one can have a postmodern atheist, and that person can pick and choose which self-refuting arguments to use.
This doesn’t seem to be a logical mistake that Christian apologists make all that often.
Grace and Peace