"Fish and chips shop" or "fish and chip shop"?

"Attributive nouns" or the first elements of compound nouns tend to be singular in form

There is no absolute rule forbidding the use of a plural noun in the first part of a compound, but it is more usual in general to use the singular form.* "Fish and chip(s) shop" doesn't seem to be an exception to this tendency. The Google Ngram Viewer suggests that both forms exist, but that "fish and chip shop" is more common than "fish and chips shop":

I don't think phonetics is an important factor

Some comments have brought up the supposed phonetic indistiguishability of "fish and chip shop" and "fish and chips shop", but I can't see how that could determine the spelling one way or another: even if it is true that nobody ever pronounces these any differently in practice (which I rather doubt), that wouldn't prevent people from using the spelling "fish and chips shop" for the pronunciation [fɪʃn̩t͡ʃɪpʃɒp]. I think the use of the spelling "fish and chip shop" is based mainly on grammar, not on phonetics.

*Some exceptions to this tendency are mentioned in the answer here: Singular/plural Nouns as Adjectives

It is a "fish and chip" shop, but you order "fish and chips".

Or just a chip shop since they always sell fish too.

In general, a shop selling a product refers to its product as a collective noun in the singular.

Chip shop — sells chips.
Cake shop — sells cakes.
Curtain shop — sells curtains.

When a noun is used as an adjective, it is almost always the singular form, even when the noun is not normally used as a singular. E.g:

• car - car alarm
• house - house key
• trousers - trouser press
• spectacles - spectacle maker

But:

• clothes - clothes line

So, "fish and chips" is the noun phrase, and becomes singular:

• fish and chips - fish and chip shop