Worked out logical implications of, in terms of anticipated experience...
Did the above and compared that to actual experience...
Evaluated for coherence (e.g., made sure that the same word isn't being used to mean two different things in the same argument)...
Evaluated for internal soundness -- that is, confirmed that the premises really do imply the conclusion, even if it turns out the premises are false...
Identified known arguments with a "family resemblance" that may form a viable reference class for evaluation or comparison...
There seems to be a hole in English (I don't know about other languages) for efficiently describing how much attention you've paid to an argument.
I can think of a variety of amounts and sorts of attention-- looks cool and plausible, examined for attractive arguments against the other side, checked math for accuracy, checked math for plausibility of application, looked for counter-arguments, looked for evidence in favor, checked factual material, checked for whether some parts are sound even if there are some errors.....
Any other suggestions for types of attention?