Appeal to Hypocrisy

Hypocrisy is the practice of claiming to have moral standards or beliefs to which one's own behavior does not conform . Calling out one's hypocrisy is fine as long as it is not used as a reason to dismiss one's argument. For example:

Televangelist X preaches about how sinful being gay is, then is caught having gay sex. People call him hypocritical. - No problem.

Televangelist X preaches about how sinful being gay is, then is caught having gay sex. People call him hypocritical, and conclude that gay sex is not sinful. - Problem

In the second example, one should call the televangelist hypocritical, then make another argument as to why gay sex is not sinful, as long as the televangelist's hypocrisy is not used as support for the argument.

Tu Quoque is a fallacy of relevance. That is, if someone's personal values (or lack thereof) are irrelevant to the argument, you cannot dismiss their argument using that evidence.

You can, however, use it to suggest someone is arguing in bad faith  after  you've proved them wrong/fallacious through other means. So if you demonstrated their argument did not work,  and  then showed they're hypocritical, you have done serious work.

The Fallacy of Inconsistency is to make contradictory claims. Something cannot be both true and not true at the same time and in the same respect. In this situation, I try to keep the focus on the person who said the statement in question. This is not difficult by asking questions, as Boghossian suggested in his book “How to Have Impossible Conversations.”