The rule is very simple, and it has no exceptions: that you add an s if you say the s, which is almost always.
That said, because we don’t say the extra s when speaking certain limited phrases like these following examples, they therefore necessarily take no added s in spelling:
- that species’ name
- this series’ final episode
- your Achilles’ heel
- Diogenes’ lamp
- for goodness’ sake
- for righteousness’ sake
(Explanation: If the word ends in unstressed /iːz/, it is invariant in the face of any /ɨz/ inflection, whether for plural or possessive forms. There are very few of these, very nearly all of which are proper nouns from Greek or sometimes Latin. The other case above is the formulaic “for ___ sake”, which has a fossilized omission in speech.)
Most speakers, however, do always say the /ɨz/ form for the possessive in all these, so they all do add a final s in writing, to show that we sake it in speech:
- that genus’s sole representative species
- Alice’s mom
- in Jesus’s name, Amen. [Note: Some people don’t actually say the added /ɨz/ for this one, in which case those people alone omit the extra s here.]
- the corpse’s decay
- the corpus’s curator
- the virus’s spread
- James’s dad
- the capital s’s shape
- Gabriel García Márquez’s greatest work
- the process’s run state
- the kiss’s wetness
- the lass’s appearance
- my boss’s idea
- the Blitz’s impact
- jazz’s exotic harmonies
- the topaz’s native color
- Aunt Agnes’s new husband
It’s important to remember that the apostrophe represents no sound whatsoever, so if you say an extra sound, you have to use an extra letter. There is no exception to this rule; it’s not as though in certain words the apostrophe suddenly stands for the /ɨz/ sound. That’s the mistake people make. If you need the sound, then you need the letter, which is s.