What's the difference between 1980s, 1980's, 1980s', 80s, 80's?
Which do I use and when?
No answer to your question will please everybody. Many people think no apostrophe should be used in your expressions:
in the 1980s
The plural genitive is formed by adding an apostrophe after the plural s, so the genitive should be:
I don't like the 1980s' music.
However, many people write:
I don't like the 1980s music.
Choose yourself. There will always be some who think you have made a mistake.
Then there are people who think it's all right to use an apostophe before the plural s if the s is added to something other than a word:
in the 1980's
With any logic, that would give us:
I don't like the 1980's' music.
I may never have seen it, though. As far as I am concerned, it is correct. English is a mess in this respect, which pretty much gives you a free hand to write it in any way you choose. You'll get complaints whatever you do. It is very common not to use an apostrophe even when the decade is written in full:
I don't like the eighties music.
That's OK with me as well. The eighties can be interpreted to be a noun used adjectivally even though it is in the plural. We can say:
the U.S. Air Force,
so why not:
the United States Air Force as well?
(If anyone wants to add an apostrophe, by all means, do!)