(Jump to the end if you just want the list.)
Let's talk about supposed grammar rules that are actually prescriptivist hogswallop. Like prepositions somehow being in a magic word class that sentences can't ever end with? “This is the sort of English," said Winston Churchill, "up with which I will not put.” You don't need to ask for whom the bell tolls. You can just ask who it tolls for.[1]
What about that who/whom distinction? My take is that “whom” has mostly disappeared but lingers on when following “to” or “for” — but even then it's optional except in certain idioms like “to whom it may concern” or “for whom the bell tolls”.
Don't get me wrong, writing constraints are amazing. I have an old Messy Matters post praising them to the sky. If you want to take up the challenge of following all the grammar rules I'm deriding here, as an exercise, akin to writing an essay with only one-syllable words or without using the letter e, by all means. You could also go the other direction. See how many prepositions in a row you can end a sentence with:
A child calls down for a bedtime story and his mother brings up a book about Australia, a topic of little interest to the child. He turns up his nose and asks, "what did you bring me that book I didn't want to be read to out of about Down Under up for?"
(See the collection of gems from whence that came. And, yes, type-of-person-reading-this-far, “Down Under” is a proper noun in that context.)
Ok, next, how about splitting infinitives? This is just faux Latin bull shorts. Say I want to say, as I did the other day, that I'm all about commitment devices as a tool to not procrastinate. Again, if I want to take up the constrained writing challenge, I could rewrite that sentence to avoid the split infinitive. But if your solution is to just force it by scooching the “to” over to the verb then you've subtly spoiled the meaning. Commitment devices are a tool for Not Procrastinating. If I say I'm all about them as a tool not to procrastinate then the reader could hear that as my commitment device obsession being about their toolishness rather than about procrastination. I mean, in this case you'd figure it out from context and cases where you wouldn't would probably feel contrived. But the split-infinitive version is still better. (Wikipedia lists other classics like “to boldly go” and “to more than double”.)
Speaking of faux Latin, what about respecting Latin plurals like “data” and “media”? This one pains me not to respect the Latin. But I have to confess that “the data show XYZ” sounds a little pretentious compared to treating “data” as a mass noun and saying “the data shows”. (Or, the former used to sound wrong? I'm actually gradually getting used to “the data show...”.) If you're talking about, say, different mediums that sound travels through, you're liable to confuse people if you insist on saying “different media", even though I take a little psychic damage from not doing so in that case. “Multiple equilibria” instead of “multiple equilibriums” is a hill I will die on though.
Ok, let's do some easy ones. The old saw about not starting sentences with conjunctions? I guess that's a useful heuristic in elementary school. And contractions in formal writing? Fine, next. Same for sentence fragments. Obviously.
If you're one of the people who were never raised in a sewer, do you need to know whether it's “less” or it's “fewer”? (To be clear, I agree with Weird Al on most of the word crimes he lists.) I guess I advocate knowing the difference, but is it worth rephrasing “you have to have less than or equal to 14 items to use the express lane”? No, it is not.
Singular they has been discussed to death; no need to belabor that one. And the prohibition on contractions in formal writing is pretty common-sensically poppycock.
I could write a whole 'nother post about commas but for now I'll just say that I'm a fan of elucutionary punctuation, where the right answer is whatever makes the prose sound best when read out loud.
In conclusion, here's my top-ten list of wrong and dumb grammar rules:
Spoiler: it tolls for thee. Womp womp.