For instance
OK, no plurals: law school entrance test
OK, head plural: law school entrance tests
?? non-head plural: law school entrances test
For instance
OK, no plurals: law school entrance test
OK, head plural: law school entrance tests
?? non-head plural: law school entrances test
Nouns that appear as modifiers in a compound noun are singular, and the need for a singular is so strong that English speakers invent singulars forms of plurale tantum words such as "jean jacket" rather than "jeans jacket".
There are certain cases in which a plural modifier cannot be avoided, mainly because a plurale tantum doesn't singularize. We cannot singularize "goods" in "goods distribution network"; if we drop the "s", we get an adjective which completely changes the meaning.
However, consider the word "supplies" which normally doesn't go to the singular word "supply" denoting one item. Yet in business we have "supply chain", not "supplies chain".
There is a strong urge to eliminate plurals from the non-head position of a compound noun; the urge only loses in cases when a special kind of plural cannot be eliminated.
If we make a noun phrase in which scissors is embedded as a non-head position, we take out the s.
We want to hear "scissor sharpening service", rather than "scissors sharpening service".
The latter is not considered grammatically incorrect, but it lacks euphony.
In some cases plural-sounding words can't be avoided in the middle of noun phrases, like "denotational semantics lecture". Semantics is a singular that people treat as a plural sometimes ("these semantics are ..."). We can't delete the "s" to make semantic, because then we get an adjective. That can be in the middle of the noun phrase, but it changes the semantics. A semantic lecture isn't one about semantics but one that has semantics (about any topic whatsoever).
Even when a plurate tantum such as "scissors" is used as a modifier in the non-head position, English speakers invent a singular form for it, giving rise to usage like "scissor blade".
The 1990 movie starring Johnny Depp could not have been called Edward Scissorshands.
(I would go as far as to suggest that "scissors blade", though widely used also, is a hypercorrection based on the forced idea that there is no "scissor".)