I am troubled by which form should I use, singular or plural, when a noun is used as an adjecitve.

The number of teachers supply needs to be curtailed.
The number of teacher suppy needs to be curtailed.
And awkward examples do not help. These are appropriate:

The number of teachers needs to be curtailed.
The teacher supply needs to be curtailed.
Students: We have free audio pronunciation exercises.
Thank you.

What I have trouble is when a plural form of a noun is allowed and a singular form of a noun is preferred.
I did a Google Book Search and seemed to have seen both "teachers supply" and "teacher supply" with seemingly no difference in intended meaning.
Sorry to use the words 'teachers" and "teacher" like an object but that wasn't my intent.
I see more singular versions -- that is versions with a singular version of a noun -- in normal situations where a noun is used as an adjective. Agree?
What do I have to know in order to make right determinations?

Why "missions" and not "mission"? I

Missions trip to third-world countries proved rewarding.
'Seemed' to have seen? Could you supply a link to teachers supply that means ' a supply of teachers'?-- at least, that is how I am interpreting your sentences. If they mean "a teachers' supply of chalk", then that is a different matter. Yes, usually the singular noun is used (as adjective) rather than a possessive form, either singular (teacher's) or plural (teachers')

I find you mission sentence odd, but you do not supply the surrounding context. Perhaps Missions is a proper name?
"...usually the singualr noun is used (as adjective)......"

How about "contract manager" vs "contracts manager"?

Is a contract manager "a manager employed on a contract basis (not permanent) regardless of their job scope" while a contracts manager is employed to handle contracts, or they both could be used interchangeably to mean a manager whose job is to handle contracts?
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The singular could be either. There are uncommon cases of plural-noun-as-modifier, and often in such cases where confusion might occur.