Conjunction "or" and the plural rule

I have been told that with the conjunction "or" we must apply the following rule: a singular verb is used if the subject that comes after "or" is singular. Is it correct?

I would naturally say the apple or the orange is good. But it seems that we should also say

• the apples or the orange is good
• but the orange or the apples are good.

Is it correct?

Otherwise, if I am thinking of "this or that or both", is it correct to use the plural? And say the apple or the orange are (both) good.

The best answer is the opposite of what you were starting to do. It's more common to use the plural and say "Apples or oranges are both good."

With mixed plurals ("they or I" or "you or we"), style guides will say to use the closer noun but there's a substantial number of native speakers who have trouble with this. The whole issue with s/v agreement and or is vexed and what we usually do is simply phrase things so that we can avoid it. If you're ever in some bizarre situation where there's a specific single orange and an indeterminate number of potential apples, you'd say "I'm fine with apples or the orange" and not anything where the sentence started with the options.

Sometimes your choice can provide context not otherwise explicit. "Is wine or beer included?" may imply belief or expectation that either none or only one of them are included, while "Are fruit, sausage, or toast included?" may belie a presumption or hope that more than just one are included.