Some would claim that these days that sentence is fine

I believe Fowler considers it acceptable now that "his" is no longer considered by some to include both males and females. But it would be better rewritten using "Authors" or "Writers" instead of "Everyone", or perhaps use the genderless "One", such as "One  should be careful to use a singular pronoun with singular nouns in one's writing."

36. Everyone should be careful to use a singular pronoun with singular nouns in their writing.

Some would claim that these days that sentence is fine, as using the singular 'they' to avoid presuming that only males have any standing in the world.

Perhaps the following sums up what NOT to do.

Everyone should be careful to avoid a male-oriented attitude in his writing. WRONG TO USE 'his'

Would 'its' be correct? Is 'Everyone' grammatically neuter or what?

William

Tips to improve your writing

Never use a preposition to end a sentence with.
Don’t use commas, that, are not, necessary.
Parenthetical words however must be enclosed in commas.
It behooves you to avoid archaic expressions.
Avoid archæic spellings too.
Don’t repeat yourself, or say again what you have said before.
Don’t use hyperbole; not one in a million can do it effectively.
Use the apostrophe in it’s proper place and omit it when its not needed.
Hopefully, you will use words correctly, irregardless of how others use them.
Eschew obfuscation.
No sentence fragments.
Don’t indulge in sesquipedalian lexicological constructions.
And always be sure to finish what

Thanks, anyway for a most interesting link result.

In the course of exploring the field, I came across the quoted list. It's a bit long, and slightly off topic - a study of Q can eventually pall after reading  dozens of comments!

From AHA! Jokes:  http://www.AhaJokes.com/

Tips to improve your writing

1. Avoid alliteration. Always.
2. Never use a long word when a diminutive one will do.
3. Employ the vernacular.
4. Eschew ampersands & abbreviations, etc.
5. Parenthetical remarks (however relevant) are unnecessary.
6. Remember to never split an infinitive.
7. Contractions aren't necessary.
8. Foreign words and phrases are not apropos.
9. One should never generalize.
10. Eliminate quotations. As Ralph Waldo Emerson said, "I hate quotations. Tell me what you know."
11. Comparisons are as bad as cliches.
12. Don't be redundant; don't use more words than necessary; it's highly superfluous.
13. Be more or less specific.
14. Understatement is always best.
15. One-word sentences? Eliminate.
16. Analogies in writing are like feathers on a snake.
17. The passive voice is to be avoided.
18. Go around the barn at high noon to avoid colloquialisms.
19. Even if a mixed metaphor sings, it should be derailed.
20. Who needs rhetorical questions?
21. Exaggeration is a billion times worse than understatement.
22. Don't never use a double negation.
23. capitalize every sentence and remember always end it with point
24. Do not put statements in the negative form.
25. Verbs have to agree with their subjects.
26. Proofread carefully to see if you words out.
27. If you reread your work, you can find on rereading a great deal of repetition can be avoided by rereading and editing.
28. A writer must not shift your point of view.
29. And don't start a sentence with a conjunction. (Remember, too, a preposition is a terrible word to end a sentence with.)
30. Don't overuse exclamation marks!!
31. Place pronouns as close as possible, especially in long sentences, as of 10 or more words, to their antecedents.
32. Writing carefully, dangling participles must be avoided.
33. If any word is improper at the end of a sentence, a linking verb is.
34. Take the bull by the hand and avoid mixing metaphors.
35. Avoid trendy locutions that sound flaky.
36. Everyone should be careful to use a singular pronoun with singular nouns in their writing.
37. Always pick on the correct idiom.
38. The adverb always follows the verb.
39. Last but not least, avoid cliches like the plague; They're old hat; seek viable alternatives.