The triage move is to find the load-bearing fallacy. Sometimes the loudest one, a colorful insult or a big emotional appeal, is decorative, while a quiet false dilemma underneath is holding the whole thing up. Ask what the argument actually needs to reach its conclusion, find the premise that supplies it, and attack the fallacy sitting on that premise.
A few patterns are worth knowing. The genetic fallacy dismisses a claim by its source, any source and not just a person. Tu quoque deflects by accusing the speaker of hypocrisy. A red herring is any distraction, usually the second move after the main fallacy is planted. For the underlying argument anatomy, see Argument Analysis.