Here is a story and useful exercise for people on the job market for junior faculty positions or postdocs: When I was a postdoc at Georgia Tech, I was asked to go through about 20 postdoc applications for my thoughts on them. This was for the Center for Nonlinear Science in the physics department. (They had already removed about half of the original set of applications, so these were ones that made the cut to still be considered.) I decided that I would try to do it in one hour. It actually took me about two, even while purposely trying to be very fast. Then, in addition to helping the Georgia Tech folks (which was a good thing to do), I asked myself what I noticed about the applications during that intentionally rushed time. I then used those insights to help me improve my application materials.
There are various ways to do such an exercise, and I suspect most people will be doing it without looking at submitted applications in the above way. Gather a set of postdocs in a sufficiently broad field (e.g., "applied mathematics" or even just "mathematics"), as it's a good part of the exercise to see how you evaluate people who are studying topics that are somewhat outside of your expertise. Make sure there is a strict time limit, as you want to see what you notice — both good and bad — about applications in that setting, as it's a realistic setting for how applications are evaluated when it counts. I think you'll pick up some good insights this way, and you can also give each other helpful advice to improve your application materials.
There are various things that we (= senior academics) talk about and give as advice all of the time, but it's good to really try it for yourself. Then you'll see what you notice — both good and bad — in the bundle of applications that you read really quickly.
And, importantly, good luck on your job hunts!
Incidentally, you can find my research statement, teaching statement, and LaTeX file (with drafts of various 'personalization' paragraphs) from my time on the job market, to get my first faculty position, over a decade ago on this Web page
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