An apostrophe used to show possession of something to someone can be a complicated thing. Do not confuse a possessive apostrophe with one used for a contraction of two words.
If you are submitting work to a publisher it is advisable to ask for their own house style as some of the examples in this article may differ from one publisher to another.
Show possession to one person or thing (singular noun)
This bit’s quite easy. If something belongs to one person or thing – to a singular noun (this includes a body of people, which is also counted as one thing) – you simply add ’s. For example:
The peacock’s tail.
We’re going to Jane’s. (indicates we are going to the place that belongs to Jane.)
The cat’s dinner.
The Green Party’s members (The members belonging to one body of people as an organisation.)
Chris’s new book.
Note the ’s after Chris. This rule is generally true of any name ending in s including surnames. Hart’s New Rules advises that if a name would be difficult to pronounce with an added ’s then drop the s: Jesus’, Moses’, Nicholas’.
Do not use an apostrophe in the possessive pronouns yours, hers, theirs, ours, its.
Showing possession to more than one person or thing (plural nouns)
If a noun is plural and ends in s, such as cats, neighbours, tables, etc. then add only an apostrophe;
The cats’ dinner. (The dinner of more than one cat.)
The neighbours’ gardens. (The gardens of more than one neighbour.)
The tables’ legs. (The legs of more than one table.)
The actresses’ scripts. (The scripts of more than one actress.)
If the plural noun doesn’t end in s (e.g. people, women, children) then the previous rule (singular nouns, above) applies and you simply add ’s:
The people’s rights.
The women’s dresses.
The children’s books.
Two people possessing the same thing
When two people are involved and they both own the same thing, let’s say Jane and Frederick have a holiday home (lucky things), the ‘s comes after the second name used:
Jane and Frederick’s holiday home. (Not Jane’s and Frederick’s.)
Apostrophes that don’t usually belong but add clarity
When writing the plural of a word that is not a noun, an apostrophe can be added for clarity of reading.
The do’s and don’ts of using a ladder. (Dos would be a confusing word to see on the page.)
A’s and i’s look very different when handwritten. (If you’d written ‘as and is…’ it could be very confusing.)
Consult your style guide
When in doubt, get out your style guide. The two I mentioned in paragraph two of this post are well-revered and accepted guides to style and you won’t go far wrong by using either, or both of them.