There are oodles of rules, some commonly accepted and others applicable only regionally or in accord with disciplinary usage. Here are a few:
Ending a sentence or clause with a quotation: The quotation marks come beforethe closing punctuation in the U.K.; they follow a period or comma in the U.S. [and I believe Canada]. If the concluding mark is a ? or !, its placement (West of the Atlantic) depends on whether the complete sentence or the quotation is a question or exclamation. For example, Did he say, “I will go”? but He said, “Will you go?” or He shouted, “I never did!” but How can you say “Never again”! In the first and third examples, the quoted matter is what calls for the ? or !; in the second and fourth, the full sentence does. (For obvious reasons, I omitted quotation marks around the complete examples.)
Single and double quotation marks: Always alternate. In Britain, the “top-domain” marks are single; in America, double, with the interior quote opposite, and a third level quote opposite that and back to the same as the top-domain use. U.S.: “Harry said, ‘Not me!’” U.K.: ‘Harry said, “Not me!”’ (And notice the placement of the exclamation point, per the previous paragraph.) U.S.: “Joe said, ‘Harry said, “Not me!”’” U.K: ‘Joe said, “Harry said, ‘Not me!’”’
Serial comma (“bacon, lettuce[,] and tomato”): Still in hot debate, but with the following consensus evolving: If writing for an editor or a teacher, follow his/her style rules. If writing for a publication, follow its preferred style manual. Otherwise, use one style consistently, and prefer usage of the serial comma for the sake of its occasional need for clarity’s sake. In general, eschew it in journalism; use it otherwise.
Phrases in apposition: Set off by commas unless extremely short and required for clarity, when they are optional: “Richard Burton, who wrote The Anatomy of Melancholy in the 1600s…” but “Richard Burton[,] the actor[,] …” In the latter use, you’re distinguishing by the short appositive that you mean the actor, not the Jacobean writer or the Victorian explorer.
Commas between subject and predicate: Classically used, following the example of German, in the 1700s; almost never used today unless needed for clarity. “The government of the United States of America[,] shall consist of a Congress, an Executive, and a Judiciary.” If Alexander Hamilton had written that, he would have included a comma; any editor or proofreader today would delete it from a contemporary essay. However: “The rule on what the appropriate use of a comma is, is dependent on circumstances.” Which is self-defining: one needs the comma separating the two uses of “is” to avoid inducing double-takes in the reader.
Apostrophe-S in Possessives: Plurals in -s (and in a few borrowed words in -x) take a lone apostrophe, with no -s following it. Singulars nearly always take -‘s with a few (often ignored) exceptions: Classic names ending in -s (and -x) traditionally take a lone apostrophe (Achilles’, Astyanax’); words ending in -ses, -sus, -sis, etc. with a /siz/ or /ziz/ sound likewise (“In Jesus’ name we pray” “Mitosis’ telophase differs from meiosis’”). The rule here appears to be whether the possessive adds an /iz/ syllable to the pronounced word or not.
There is of course much more: semicolons, avoiding comma splices, etc.