Both words often work for possibility: I might be late and I could be late mean much the same, though might can sound slightly less certain. But when ability or a real opportunity is part of your meaning, you need could. You could try emailing her suggests a way to act; she might fix it cannot. A suggestion is softer ground: since weighing an option is itself a kind of possibility, some writers happily write you might try emailing her, and it reads as a gentle nudge. Could simply keeps the sense of a workable option more clearly in view.
When either word fits, choose by tone
I might join you later versus I could join you later. Both are natural in a message. Might keeps it neutral, while could leans toward an offer, as if you are floating the idea for the other person to pick up.
We might reschedule if the storm hits versus We could reschedule if the storm hits. Might reports the chance from the outside, almost like weather commentary. Could puts the decision partly in your hands, so use it if you actually want to propose rescheduling.
She might have missed the email versus She could have missed the email. Both speculate about what happened, and in a quick note either reads fine. Could have can add a hint that she had the opportunity, so the nuance matters more when that distinction affects your next step.
For possibility messages, the two words usually overlap, and either choice sounds natural in everyday notes. Might tends to sit a touch more measured, which suits messages to someone senior or any reader you want to keep at a gentle distance. In quick notes to teammates, pick whichever one sounds like you.
One situation shifts the decision noticeably. When your meaning includes ability or a way to act, could is the natural choice, and most writers reach for it when offering advice. You could try emailing her offers a concrete step; you might try emailing her mostly reports a vague chance, and most readers take it that way rather than as advice. The same pattern explains why she might fix it cannot suggest a fix the way she could fix it does.
Quick check: you want to tell a colleague the meeting may run over. Type We might run long and We could run long, then read each aloud and notice which one matches what you actually want the reader to do.