"Car sale" vs "Cars sale"

Solution 1:

In your examples, car and rule are noun adjuncts: nouns used as adjectives. Wikipedia notes:

Noun adjuncts were traditionally mostly singular (e.g. "trouser press") except when there were lexical restrictions (e.g. "arms race"), but there is a recent trend towards more use of plural ones, especially in UK English. Many of these can also be and/or were originally interpreted and spelled as plural possessives....

Car sale and rule extraction are both traditionally correct. Rules extraction fits the recent trend, but cars sale would be problematic: cars' sale (the selling of cars) doesn't mean the same thing as car sale (a discount on cars).