1. To boldly go where no man has gone before.
This line (and its newer version, with no one in place of no man) is as famous for being "wrong" as it is for being from the intro to each episode of Star Trek.
What's "wrong"? It's a "split infinitive," with boldly improperly between to and go.
Is it really wrong? No. The "rule" against split infinitives is just a grammatical superstition. It was invented in the 1700s by a grammarian who wanted to "improve" the language along Latin lines. English, however, is not Latin, and the option of putting words between to and the verb root has always existed and has often been made use of by respected authors. There are times when a sentence works better if you don't do it, sure; that doesn't make it a rule, and the Star Trek line is not one of those times, either. "Boldly to go"? "To go boldly"? No.
2. The woods are lovely, dark and deep. / But I have promises to keep, / And miles to go before I sleep, / And miles to go before I sleep.
This is from "Stopping by the Woods on a Snowy Evening," by Robert Frost.
What's "wrong"? It starts a new sentence right before a conjunction ("But...").
Is it really wrong? No. It has always been an option in English to start sentences with conjunctions, and the most respected authors in the language have done so on occasion to good effect.
3. This was the most unkindest cut of all.
Marc Antony says this in Shakespeare's Julius Caesar.
What's "wrong"? Double superlative: most unkindest.
Is it really wrong? It is now, but it wasn't when Shakespeare was writing. What happened between now and then? Those grammarians of the 1700s set out to "improve" English. Many of their "improvements" didn't really stick, except as superstitions, but the laws they laid down against double negatives, double comparatives, and double superlatives did.
4. We are such stuff as dreams are made on; and our little life is rounded with a sleep.
Prospero says this in The Tempest by William Shakespeare.
What's "wrong"? Ending a sentence (or a clause) with a preposition ("on"), and using a semicolon between two conjoined clauses.
Is it really wrong? The "rule" against ending sentences or clauses with prepositions is another fake rule, invented a century after Shakespeare died. It has been ignored to good effect by respected writers throughout English history. Just try rephrasing the line to avoid it without making a mess. The semicolon is an error in the standard English of our time, but the rules of punctuation were looser in Shakespeare's time.
5. Well, back to the old drawing board.
This is from a cartoon by Peter Arno showing a designer walking away from a crashed plane.
What's "wrong"? This sentence has no verb.
Is it really wrong? As covered in "It's totally okay to write incomplete sentences," there are sentences that can use an adverb as the predicate rather than filling extra space needlessly with added verbs. It's usually a bit less formal, but it's misguided to say it's actually wrong. Many an erudite reader has read "And so to bed" in the diary of Samuel Pepys without emitting a peep about it — and that begins with a conjunction too.
6. The game isn't over till it's over.
This is attributed to Yogi Berra.
What's "wrong"? Tautology; use of till.
Is it really wrong? Logical tautology — the game is over when it's over, obviously — is not a grammatical error, and it can be quite useful as a rhetorical device, because it trades on the different uses of a word. Would you rather Berra had said "Do not believe at any point that the remainder of the game is a foregone conclusion"? As to till rather than until or 'til, that is in fact not an error at all; till is even older than until, and is still accepted in common use. In fact, until was formed from und (an Old Norse word meaning "as far as, up to") plus till. Since till means "up to" (it's different from "till the soil" or "cash in the till"), until is actually a bit redundant.
7. Frankly, my dear, I don't give a damn.
Rhett Butler says this to Scarlett O'Hara in Gone with the Wind.
What's "wrong"? Not the damn; profanity is not ungrammatical, just impolite. No, it's the frankly, because he's not saying "I don't give a damn frankly" as opposed to giving it some other way. The best-known infraction of this "rule" involves the use of hopefully.
Is it really wrong? No, not at all. Adverbs used at the beginning of a sentence to describe the speaker's attitude are called sentence adverbs and have been used in English for a very long time: frankly, obviously, really, clearly, sadly, seriously, obviously, hopefully…
8. A well-regulated militia, being necessary to the security of a free State, the right of the people to keep and bear arms, shall not be infringed.
This is the second amendment to the Constitution of the United States of America.
What's "wrong"? It has commas in places where they are not allowed by the rules of current English.
Is it really wrong? It would be if they were writing it today, unless the core sentence were "A well-regulated militia shall not be infringed" and the other clauses were appositives — but that's not the standard reading. As long as the subject of the verb is "the right," the problem is that there's a comma between it and the verb "shall." Also, the comma after "militia" appears to be in the middle of a clause — but there are disputes about that too. The problem is that rules for commas were a bit looser at the time. It's not that the Founding Fathers didn't know English; it's that they didn't know what it would become more than two centuries later.
9. You ain't heard nothin' yet, folks.
Al Jolson said this in the first talking motion picture, The Jazz Singer. He probably wasn't the first to say it and he definitely wasn't the last.
What's "wrong"? Use of ain't, use of a double negative, and "dropping the g."
Is it really wrong? See above regarding double negatives. "Dropping the g" is an inaccurate way of referring to the fronting of the "ng" sound to "n" (there is no "g" in the sound; we just write it with g because we don't use a nice single standard letter such as ŋ for it). This was in fact the standard way of saying –ing for quite a long time and is reflected even in respected poetry from the 1600s and 1700s. But schoolteachers in the 1800s started teaching a spelling pronunciation, and you know how it is with things you're taught in school: even if you don't do them yourself, they're burned in your mind as "correct."
As to ain't, this is a word that now exists in English especially for the purpose of being "wrong," informal, slangy. It first came about in the 1700s and 1800s from contractions for am not, are not, and have not. It was commonly used in casual speech at many levels of society for a time, and by educated speakers in some areas as late as the turn of the 20th century, but prescriptive grammarians didn't like it and banned it. Once that happened, it secured its place as a distinct marker of pointedly nonstandard speech. You may feel certain that Al Jolson said what he said precisely because it was not standard English. Can you imagine a jazz singer saying, "You haven't heard anything yet?" Where's the groove in that?