My boss makes grammatical errors in his emails and written reports that make him look foolish. Should I correct him?
Calling your boss' spelling blunders and grammatical lapses to his attention isn't a good political move, since he can't correct or undo anything he has already written. Instead of eliciting his cooperation, you might just end up making him feel embarrassed, and putting him on the defensive.
So I would suggest a more positive, less particular, approach that won't appear to be aimed at his shortcomings in written expression. Point out that you have noticed a continuing pattern of mistakes in company communications, from reports to emails, and that you are concerned those types of errors may tarnish the company's image. If he asks for details, be prepared to provide general examples without naming the culprit, i.e., him. Now that you've identified the problem, you can quickly offer a solution: Let him know you have a knack for good grammar, and would love the opportunity to exercise it. Tell him you would like to keep your skills sharp by working on the very documents that can adversely affect the image of your company as well as its leaders.
This way, rather than singling him out, you've offered a fix for a problem that won't cost him anything, including his dignity. Chances are, he'll be impressed with your willingness to serve the team, and welcome the chance to make it shine even brighter.