It can even influence a language: Fetus is the original Latin term, but it got "fixed" to foetus or fœtus both in how people wrote Latin, and also in languages that had a term derived from it so now foetus is sometimes found in English, Dutch and German and also fötus in German; all based on people "fixing" what wasn't broken.
In between personal hypercorrection and something introducing a new spelling that became common across four different languages, are myths like "the passive voice should be avoided", "don't start a sentence with a conjunction", "don't start a sentence with however", "don't split infinitives", "don't split any verb form", "don't end a sentence with a preposition", "don't use the genitive of an noun that represents a non-living thing", "don't use like as a conjunctive". Here not only has someone got a bizarre notion into their heads about what is correct, but the "rule" has ended up being actively taught.
Often it's combined with an incorrect understanding of its own terms, as in this notorious example:
Not only is this teacher enforcing an imaginary rule, but of the 10 cases marked as "P.V.", only three are in the passive voice. Of those 3, only one seems like it could be rewritten into something that didn't weaken it (and that one still wouldn't actually improve). One actual use of the passive voice isn't marked as such.
In theory, such a "rule" could become so popular, that it became a real rule just by dominating the thinking of those who spoke the language. Many though are actually impossible to follow consistently, or just too at odds with common literate use, to ever reach that point.