Method 1: Disagree with one of the premises (subtype- neither accept nor reject the premise, but argue its truth is unknown or unknowable)
Method 2: Disagree with one of the inferences (subtype- neither accept nor reject the inference, but argue its validity is unknown or unknowable)
Method 3: Show that the argument implies something else known or suspected to be false, indicating that one or more of the premises or inferences must be wrong. (Special case, show that the argument leads to a contradiction)
Method 4: Present an argument you claim is stronger for the opposite conclusion without considering why the original argument is wrong.
Method 5: Play Moorean inversion- My belief in the opposite of the conclusion is much stronger than my belief in the premises of the argument, ergo at least one of the premises or inferences must be wrong. (As Bob Churchill notes in the comments, this is the special case of 3, in which ‘the thing implied known to false’ is the conclusion itself).
Method 6: As noted by the commenter N3: - “arguing that the interpretation of a premise is ambiguous between one possibility which is true but doesn't lead to the conclusion vs. another that is false and leads to the conclusion.” At first I thought this is best treated as a case of method 1 or 2, but I now suspect the overlap is awkard enough it deserves its own category.
You forgot parody arguments. They show that the form of argument fails even if you can’t identify the mistake
Method 7: Attack the author of the argument for being an irredeemable cur, worthy only of insult and ignominy.