How we change what others think, feel, believe and do
Given an argument with which you disagree, you can mount an attack on it in a number of ways.
In a formal argument, the primary arguer must establish a prima facie case (that stands on its own) and thus carries the burden of proof. The opponent only needs to show that the case is not proven to win the argument and thus may well focus on attacking and disproving the given case. An alternative case may also be given, but is not needed.
Refuting is also known as rebuttal, the burden of rebuttal or the burden of clash.
Look at the words used in the argument. Is their meaning clear? Is there one meaning only for each? If you can detect vague meaning or ambiguity in the parts of an argument, then you can show the whole argument to be shaky -- and, of course, you can shake it until it collapses.
Consider the rationale being used. Test each statement for logical soundness. Also test between statements across the argument.
Dig into the data and evidence being used to support the main claim.
Look at the supporting statements to the argument. Seek cracks and chinks in the armor. Look for a place to drive in a wedge. Many arguments have a valid claim but weak support.
Create another argument that uses more correct logic, that is more powerful and all-encompassing than the given argument.
And of course you can use fallacies of your own, of which there are many. This, of course, may be refuted itself. So consider your audience and whether they are capable of such refutation.