Before I discuss the possessive form (or case), let’s review a few grammatical terms and rules.

A compound subject has two or more subjects:

Susan, Terri, and Samantha share a flat.
Carrots and beets are root vegetables.

Similarly, a compound object has two or more objects:

I will call the director and the assistant director tomorrow.
I ate nuts and berries for breakfast.

Compound nouns (lampshade, footstool, housewife, cupboard, real estate) are formed by combining two or more words. (Usually at least one of the words is a noun.)

Some compounds are open (there is a space between the words: ice cream, school bus, life jacket, middle class, hot dog). Others are hyphenated (six-pack, two-year-old, well-being, mother-in-law). Closed compounds (football, birthday, bedroom, notebook) omit the space or hyphen. The spelling can remain in flux for a long time as a language evolves, with disagreement about which version is all right (alright?). Also, as conventions sometimes differ by region, profession, and country (e.g., the spelling of color/colour in the US/UK), you may see northwest as well as north west, and northwestern as well as north-western.

A plural noun is formed in different ways; the simplest is by adding an s:

books
germs
concepts

Exceptions abound, and other rules may apply if the noun ends in ch, f, fe, o, s, sh, ss, x, y, z, us, is, or on (e.g., witches, thieves, wives, gases, dishes, misses, oxen, parties, quizzes, viruses, crises, criteria).

For words ending in o, -es is often added to form the plural (heroes and potatoes) but not always (pianos) and, often, either form is acceptable (stilettos, stilettoes). Then there are the words that stay the same whether they are singular or plural (sheep, series) and the irregular nouns (foot/feet; woman/women). (A complete discussion of this topic is beyond the scope of this post.)

Don’t confuse the plural with the possessive!

The witches hatched a plan. (plural)
The witch’s brew was delightful. (possessive)
The witches’ plan backfired. (plural and possessive)

Use the possessive form to indicate ownership or possession:

My tooth aches.
Harry’s job is difficult.
Linda’s husband is out of town.
His boss’s schedule is insane.
Your daughter’s performance was impressive.

For most singular nouns, the possessive is formed by adding apostrophe s (‘s). The same is usually true for nouns referring to groups of people (states, countries, organizations, and associations):

New York’s housing crisis is worse than Boston’s.
Georgia’s governor declared a state of emergency.
England’s currency is the same as Scotland’s.
The school’s future is in doubt.
The company’s profile can be found on the website.
The firm’s clients include celebrities and politicians.

Buildings and objects are not considered “owners” in the same way that people and pets are. Compare:

The dog’s leg is broken.

I’ll meet you in the hotel lobby. (not “the hotel’s lobby”)

The view from the restaurant is spectacular. (rather than “The restaurant’s view is spectacular”)

As with the last example, above, a sentence is sometimes best rewritten to avoid using an apostrophe s:

The exterior of the building is being restored. (instead of “The building’s exterior is being restored”)

No definitive rule governs inanimate objects, however, and some constructions using apostrophe s are common and acceptable.

The car’s features include heated seats and a navigation system.
but not
Don’t slam the car’s door! (car door)

Thus, you are likely to encounter both:

The terms of the agreement clearly prohibit such transfers.
and
The agreement’s terms clearly allow such transfers.

The wine’s flavor will be affected by its age.
and
The color of the wine remained unchanged.

For plural nouns, only an apostrophe is added when the noun ends in s:

the secretaries’ strike
the workers’ plan
the musicians’ concert

When the plural noun does not end in s, as in people or children, add an apostrophe and an s:

people’s court (not peoples’ or peoples’s)
children’s playground (not childrens’ or childrens’s)

For hyphenated compound nouns, an apostrophe and an s are usually added at the end of the word :

Tomorrow is my son-in-law’s birthday.
but
Today is my two-year-old son’s birthday.

If you need to indicate possession for a plural compound noun (e.g., daughters-in-law or attorneys general), you will usually be better off with “The group of daughters-in-law was created to provide support for women with difficult in-laws” instead of “The daughters-in-law’s group was created to provide support . . . “

Similarly, instead of:

The annual attorneys general’s conference has been scheduled.

write:

The annual conference of attorneys general has been scheduled.

(Note that a conference of generals would be a “generals’ conference,” but the plural form of attorney general is attorneys general. A group started by your daughter-in-law would be your “daughter-in-law’s group.”)

Forming the possessive with a compound subject or compound object

Question: If Tim and Joe are brothers and you are going to visit them, do you say, “I’m going to Tim’s and Joe’s house” or “I’m going to Tim and Joe’s house” or “I’m going to Tim’s and Joe’s houses”?

Answer: If Tim and Joe live in the same house, only one apostrophe s is needed. (Tim and Joe’s house)

If Tim and Joe live separately and you are going to two houses, the possessive form would be applied to each of them. (Tim’s and Joe’s houses)

Try substituting their for “Tim and Joe”: “I’m going to their house” (if they live together) or “I’m going to their (two) houses” (if they live separately). In common parlance, people often say things like, “I’m going over to Tim’s and then I might stop by Joe’s place.”

In the sentence “Trish walked to Tim’s store and Joe’s house,” Trish is the subject, walked is the verb, to is a preposition, and the compound object is Tim’s store and Joe’s house. The apostrophe rule is the same for compound subjects. (Trish’s hair and Tina’s makeup looked great.)

John and Mary’s religion is really strange. (John and Mary share the same religion)

John’s and Mary’s beliefs are very different from mine. (John’s beliefs and Mary’s beliefs are not the same; their beliefs differ from my beliefs)