I am looking for strategies for catching mistakes in graduate exams eg. qualifying.
The more people suggest, the better because what is obvious to you, might not be to others.
Most of the advice in this site and others is about computational mistakes. But my most usual mistake is writing down a conceptual proof I thought is correct, only to later find it is not. Had I found it, I would have worked out the correct one.
Here are some strategies of mine
"I have found that the best way to cope is to change my attitude. First of all, when I get an answer to a problem, I do not assume the answer is correct. I assume, rather, that I did, in fact, make a mistake and am now in the editing phase of the problem solution. How one edits depends on the problem to be solved. "
Test your proof with an example. Draw pictures.
Did you prove something stronger than you were supposed to? Draw any conclusions from your method of proof. Do you get any contradictions?
Condensing a proof, forces you to understand it better and thus to find mistakes.
As you are preparing for the exam share your practice-work with others, so they can catch your mistakes.
6) Write in detail
When coming up with proofs, write them in detail so that the mistakes become more obvious (compared to when holding a proof in your head while checking for mistakes).