Question: I’m a big fan of music from the 1960s. I’ve seen it written a lot of ways: The ‘60s; the 60s; the 60’s.; even ‘60s’ music. Which one of those is correct?
Answer: You’ve treaded here into one of the stickier grammar rules: apostrophes used to indicate plural nouns – though, in your example, the rule would be misapplied. We’ll get to that in a bit, though.
To answer your first question, if you’re talking about a specific decade like the 1960s, and you want to be a little less formal, writing it as “the ‘60s” is perfectly fine. There the apostrophe indicates an omission – specifically the omission of “1900.” It’s the same principle applied in some of the more cringe-worthy slang you might pick up from a Mark Twain protagonist: ‘cause (because), s’pose (suppose), etc.
Now let’s go through the rest of your examples, shall we? Let’s say you want to turn a number into a plural object. To stick with the 60 theme, let’s say you’ve got a very large group of people which you’ve counted off into 60 smaller groups. If you’d like each member of the number 60 sub-group to, let’s say, line up for lunch, you might say this: “Would the 60s please come to the cafeteria?”
The principle applies for all numbers (7s, 18s, 4,000s, etc.), and also for capital letters (As, Bs, Cs, etc.).
Of course, since nothing can be too easy in English, if you’d like to make lower case letters plural, you’ll need to bring that apostrophe back. Remember when your mother advised you to mind your p’s and q’s? She used apostrophes to do so.
The third 60 you’ve presented here, “60’s” is simply incorrect … unless of course you’re talking about a possession belonging to, I don’t know, a robot named “60.” But since robots still don’t have personal property rights, it’s not something we need to worry about.
The last 60 in your question, “’60s’ music,” is, I think, a valiant attempt to apply correct grammar. Ultimately, of course, it is one that fails. The initial apostrophe is used correctly to indicate that something has been omitted (“1900”). The second apostrophe wants to, I believe, indicate that the music in question comes from the 1960s. I can understand this. Why it is incorrect is this: The music does not belong to the 1960s; it is simply of the 1960s. In this use, ‘60s becomes an adjective in the same way you might say “Jamaican music” or “Celtic music;” thus no apostrophe is needed.