The question of how and when to use the possessive S versus the plural form in English can be confusing.
In the examples above, we can quickly eliminate the option of “doctors appointment” because in that phrase, the noun “doctor” is acting as an adjective to the word “appointment” and adjectives are never made plural in English.
Think of the phrases “car factory” or “basketball court.” The words “car” and “basketball” are both nouns acting as adjectives to give us more information about the factory and the court. We would never say the “cars factory” or the “basketballs court.” So, we shouldn’t say the “doctors appointment”, either.
We could, however, say “a doctor appointment.” This, though, is almost never used or heard.
The first example given of a “doctor’s appointment” is the correct form of the possessive and used appropriately here. You may be wondering why the possessive would be used here since the appointment seems to belong to you and not to the doctor.
Yet, if you take the time to consider it more carefully, you can see that the appointment does indeed belong to the doctor. When you book an appointment with the doctor, you are taking some time out of his daily schedule. He has appointed a period of time to see you. Thus, the appointment truly belongs to him and as such you should use the possessive S to communicate that using correct grammar.
Just remember that we always use the possessive S to show possession.