There are a number of grammatical rules that are the subject of some level of dispute. The modern trend is to dispense with prescriptivist rules (some would call them "grammar myths"). For example, "don't start a sentence with a conjunction," "don't end a sentence with a preposition," "don't use contractions," "don't write in the first person," "use the Oxford comma," "hyphenate in cases such as 'fifty-page report' or 'twelve-member jury,'" "never use "their" as a gender neutral third person singular pronoun," "put a double space before the beginning of a sentence" and "don't split infinitives."
The issue here is that some people consider it to be improper to neglect these rules in formal written English, and it may cause a reader who is narrow minded to view you as ill trained in grammar.
My own inclination is to break these "soft rules" only consciously and with the audience in mind. Where communication per se is paramount, or stylistic effect is enhanced, one may go ahead and break them, but one should not be habituated to styles that violate these rules outside a firm or author specific style manual, if one is to resolve recurring issues optimally.
"don't start a sentence with a conjunction"
I routinely start sentences with "but," which is short and usually can replace wordy and less clear constructions. I avoid frequently starting sentences with "and" unless there is good reason for it, since some readers will get the impression that it is incorrect, or because it gives the appearance of writing with sentence fragments and concealed run on sentences.
"don't end a sentence with a preposition"
Similarly, while ending a sentence with a preposition is perfectly acceptable in spoken English and in informal written English including fiction, doing this frequently conveys the wrong impression in formal written English because many people have internalized the "grammar myth" that this is improper.
"don't use contractions"
Contractions are likewise common place and perfectly acceptable in almost every form of English communication except formal written English, and even then, aren't entirely unacceptable. But, because of the "grammar myth" that disfavors the use of contractors in formal written English, contractions should be used more sparingly in these contexts.
"don't write in the first person"
One stylistic preference that I do observe in formal written English, with rare exceptions where I rely solely upon my personal knowledge of a specific fact, and also observe when possible in formal spoken English, is that of not writing in the first person.
There is often a reason to avoid the first person in formal writing or formal speaking, despite the trend to do otherwise. It is psychological. Removing a reference to the author in your writing, psychologically, takes the author's credibility out of the reader's mind discouraging ad hominem attacks to what is said. Instead, it creates the impression that a statement is a fact has been made by the author.
The movement to return "I" to formal academic writing is largely a political outgrowth of identity politics. One of the axiomatic premises of this movement is that the meaning, relevance and force of a statement is intimately intertwined with the identity of it author. And, there are indeed circumstances when knowing something about the author's identity can color a statement's meaning. But, the axiom is simply not a generally true statement. Statements are perfectly capable of having clear meaning, of educating, of conveying information, of persuading listeners, even in the absence of any knowledge of the author's identity. Often, the author's identity is irrelevant. More often than not in formal acadmeic writing, inserting one's identity into the discussion undermines, rather than enhances, an argument.
"use the Oxford comma"
Oxford commas (a comma before the word "and" in a list) can sometimes have a semantic meaning, but can sometimes look slightly pedantic. They are not universally used in good formal English writing, where a contrary stylistic trend favors grammatical minimalism. Unless necessary for clarity, I disfavor the Oxford comma.
"hyphenate in cases such as 'fifty-page report' or 'twelve-member jury'"
The modern trend is to disfavor hyphens, that are not absolutely necessary to avoid a plausible ambiguous construction, unless it is part of a proper noun. Hypens are ugly. Compound words, or pairs of words with a particular meaning when written together, are more accepted. Most hyphenated words are in transition and disputable in any case. In practice, omitting a hyphen rarely sacrifices clarity. But, not everyone agrees.
There are a few specific exceptions, generally involving cases where a hyphen connects syllables rather than words that can stand on their own (e.g. "co-ed"), although the trend in these cases is towards fusing words without hyphens, or near universal hyphenation practice (e.g. "Asian-American").
In any case, hyphenation is not a matter that can be resolved with a single rule. Each particular instance must be considered individually and hyphenation practices are among the most rapidly evolving parts of modern English grammar.
"never use "their" as a gender neutral third person singular pronoun"
The use of "their" in spoken English as a third person singular possessive can be tolerated. "A teacher who fails to plan does will not achieve their goal."
But, in formal written English, it is usually better to rewrite a sentence to avoid the dilema. If a third person singular possessive in a context intended to apply to both genders cannot be avoided, either "his" or "her" can be acceptable, with the gender more likely to apply included.
"put a double space at the end of a sentence"
I always used to put a double space at the end of a sentence because I was told that it was a rule and never questioned it. It has become a habit. I've seen convicing arguments for the contrary rule, that one should never use a double space in written English. But, old habits die hard. If I could overcome my old habits consistently, I would abadon this rule. But, since I can't, I prefer consistency even if this means adopting an anchronistic spacing rule.
"don't split infinitives"
In Latin, infinitive verbs are created by inflection, so it is impossible to split an infinitive. In English, an infinitive such as "to be" or "to go" is a two word construction, so it is possible to split an infinitive. One of the classic examples of the acceptability, "to boldly go" illustrates that there is nothing informal or jarring about a split infinitive in English. Ignore this "rule."