The first two are usually pluralised

Number three requires to have something else you do not like

Number 4 also requires something else you don't like

both either and nor require a choice between two or more things,

either is a choice between two things and neither is a choice between two or more things.

Neither requires to come before the choice of two or more thing or to introduce a negative statement

I do not like X nor do I like Y [eliminates the neither that requires two choices follow it]

I do not want to have anything to do with A nor do it wish to see either B or C

Here it is explained by a dictionary more simplified than any explanation you will find else where.

1. I neither like tomatoes nor potatoes or plums.

2. I like neither tomatoes or potatoes nor do I care for celery .

3. I don't like potatoes or tomatoes either [note the or between the two very essential]

4. I don't like tomatoes nor potatoes neither [note the necessary nor between the two you can argue that the nor is redundant but it is the only way to resolve your bad logic]

4a. I don't like tomatoes neither do I like potatoes or grapes.

notice nor that equals [not or] has two or more choices following it and either has only one choice.

to link together more than two choices or statements with either you must use "and" or "or" as a conjunction the joining together word.

use nor to simplify and remove "neither" being required if there is only one choice OR change the sentence to use "either".

here is a beginner level quiz for you to try

it is best not to use either neither and nor until after you have had extensive tuition from a component teacher/tutor it can get very complex and involved. It is the basis of artificial intelligence get it correct and you can do wonderous things get it wrong and you can make a mess.

1. I neither like tomato nor potato. Wrong

2. I like neither tomato nor potato. Correct

3. I don't like tomato either. Correct

4. I don't like tomato neither. Wrong.

Sentence #1 is wrong because "neither like tomato" doesn't make sense with "nor potato": if you "neither like tomato", then we expect "nor [another verb] tomato"; you should be talking about two different things you don't do with regard to tomatoes.

Sentence #2 is correct. In this context, using the singular words "tomato" and "potato" without any article is one way to make a general statement about tomatoes or potatoes. However, this usage is less common today that it was in the past, and saying "I like neither tomatoes nor potatoes" is more common today.

Sentence #3 is correct.

Sentence #4 is wrong, because we don't use "neither" when the verb is negative.

I think potato and tomato should be cast in their plural forms, potatoes, tomatoes, to start. It sounds rather odd to me to say one does not care for a single potato or tomato when the context is clear that a plural is meant.

With casting the nouns in their plural forms, sentences 1,2 and 3 are fine but 4 is not as it is a double negative.

neither......nor always go together although it is a very common error of native speakers to say neither...........or.

In my opinion

Sentence one is common to say

Sentence two not so, but still right

Sentence three is alright

Sentence four I would say wrong.

Hope this helps!