The first example is about one particular year, 1960, while the rest of the question seems to be about decades. The 80s is, roughly, the ninth decade, not the eighth.

"in their 80s" means their ages are 80-89.

"1960's chart listings" and "1960's fashion" might be considered saxon genitives, but I'm not sure. They could also be viewed as inversions of "chart listings of 1960" and "fashion of the 1960s". We don't consider years to be "owners" of things that happen during them.

I consider most appearances of 's with years to be erroneous; it's not something that belongs to 1960; it's a characteristic of a collection of ten years beginning with 1960.

Don't use the possessive / Saxon genitive to reference 1960's music (or 1960s' music). The word 1960s is a plural noun adjunct, as with the singular noun adjunct jazz music ("jazz" being a type of music - a noun, not an adjective).

The gap between plural-form attributive usages and associative rather than true possessive genitive structures has become blurred. Thus "working mens club/clubs" is often encountered, along with "dogs home/homes", "travellers cheque/cheques", "writers guild/guilds", and the similar-looking "St James Park" (other variants, with apostrophes, exist for different examples of this one). Look up in a Google search how many instances of say "a 1960s pop group" are afforded a hyphen.

This question is similar to: Apostrophe Use In Abbreviated Decade Which Is Also A Possessive. Sven Yarg's answer addresses also the use or not of the apostrophe in 'possessive' cases, such as << the 1990s' main contribution to popular culture ... >>. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem.

Yeah, but the Beatles' songs is de rigueur.

@EdwinAshworth They specifically said "Whether or not to use the Saxon genitive apostrophe is not my question."

@FumbleFingers What about "1960's chart listings", which is about the single year 1960, not the decade?

@EdwinAshworth - are you saying that the 1960s (i.e. the 10 years of that decade) is not a Saxon genitive?

@Barmar: I know we're on ELU, not ELL (and it's a well-posed question, Dan). But I think "1960's chart listings were radically different to 1950's" is just a clunky way of saying "The chart listings of 1960 were radically different to those of 1950". And let's not forget - in speech, you couldn't tell those apostrophes were there anyway, so the meaning might have been "The chart listings of the 1960s [decade] were radically different to those of the 1950s".

Anyway, I refer both of you guys to this comment thread re "pathological" cases..

@FumbleFingers The original sentence said "Chart listings in the year 1960". Then they inverted it to "1960's chart listings". I realize that in speech you can't tell whether this refers to the year or the decade, but I'm going by the OP's intent.

Whatever the intent was, the reality is it's a dead end exploring the limits of natural language - it's not like refining to the nth degree how exactly the most obscure construction in a computer language should be implemented by the compiler (where we require precision, and can't tolerate ambiguity). English just doesn't have the flexibility to handle certain situations unambiguously using the kind of shorthand syntax that works 99.99% of the time. And it ain't worth bothering for that 0.01% case - just use far more words until you've made the meaning clear.

@Barmar The answers address all the usages, attributive (including plural-form) and use of apostrophe in pure plurals (discouraged here nowadays). According to Sven Yargs' answer in the duplicate, the apostrophised form is reasonable in "the 1990s' main contribution to popular culture ..." (when of course it is the Saxon genitive). But not in "the 1990s were a time of ...".

@Barmar - thanks for noticing and emphasising that the apostrophe is not the point of this question.

@EdwinAshworth - are you saying that the 1960s (i.e. the 10 years of that decade) is not a Saxon genitive? >> "June Forsyte was a character in the series" / "The Forsyte Saga" // "Great music was written in the 1960s" [not usually spelled with an apostrophe] / "The 1960s era was in complete contrast to the 1950s": no genitives involved. //// But "the 1990s' main contribution to popular culture" is, as Sven points out at the original, the preferred variant and obviously a Saxon genitive. Compare "the Beatles' main contribution ...".

@EdwinAshworth - I think we may be at cross-purposes. Although I generally DO use an apostrophe for a Saxon genitive (and I try to place it correctly) I also accept that this apostrophe is often left out. My question is NOT about the apostrophe. It is whether, in phrases like "I'm in my 60s", "the Beatles pursued solo careers in the 1970s" the numbers are examples of Saxon genitives (where familiarity has resulted in the noun adjunct on its own being sufficient).

The definition of Saxon genitive is given by Wikipedia, YourDictionary, and many other sources: << For nouns, noun phrases, and some pronouns, the possessive is generally formed with the suffix[/clitic] -'s, but in some cases just with the addition of an apostrophe to an existing s. This form is sometimes called the Saxon genitive. >> The apostrophe is non-optional in such examples. If you wish to use the term in an unusual way, this must be made clear, and a supporting reference given.

... Perhaps you are actually enquiring about the '[partial] demise of the apostrophe' covered say at Is it correct to say I write children books (ie not in the possessive case?, and travellers cheques / customers requirements .... Here, apostrophe-less travellers cheques and writers guilds and the like are discussed

@EdwinAshworth - thank you, the first of these two links is very interesting!

@FumbleFingers - "1960's chart listings did not include Beatles songs" is surely less 'clunky' than "The chart listings of 1960 did not include Beatles songs"? I agree that "...in speech, you couldn't tell those apostrophes were there anyway, so the meaning might have been "The chart listings of the 1960s [decade] were radically different to those of the 1950s". THAT is precisely my question - which do you hear?

Who cares what any one person "hears" when reading effectively "contextless" text (where we can't even be sure whoever wrote it knew exactly what he was doing when he chose whether and where to put an apostrophe? If you want to be unambiguous, choose different phrasing

Do you perceive a difference between "1960's events" (the events of 1960) and "1960s events" (the events of 1960-69)? Are you unsure if they're the same? Do you want to know the difference? Asking "Is this an example of X?" isn't a good question if you don't know what X is, and it's not clear what you mean by "is this a Saxon genitive?" because on one level it's clearly not.

@StuartF - curious, that's all.