Real grammarians, i.e. linguists who study the grammar of English as it is, teach us that these aren't actually rules of grammar anyway, so much as prescriptions that were made up out of whole cloth for various reasons and that never had much to do with the way English was spoken or written.

But do also note that a lot of people do believe those prescriptions to be valid, and view breaking them to be low status. All the "singular they is fine" blog posts in the world are irrelevant if using singular they will annoy half your readers.

Of course, I tend to use singular they anyway. It's often the best alternative and I doubt that many people in my likely target audience will really care. But you should still know the biggest things that will annoy people, so that if you use them, it will be out of conscious choice and not of ignorance.