With regard to the expression on a “wild goose chase” does it mean chasing wild geese or wildly chasing geese? Does “wild” modify goose or chase?
Shakespeare, again. Looks like it originally was related to a type of horse event where the horses chased after the lead horse at a set distance, mimicking geese flying in formation.
Chasing wild geese would indeed be a fruitless endeavor. I suspect that it was one of those colourful phrases like the more modern “herding kittens”.
That ambiguity is common in English. Compare “Big Garage Sale!” Is that a big sale of garages? Or a sale of big garages?
English has the peculiarity of having “adjunct nouns”, which is a noun functioning as an adjective to modify another noun, resulting in a noun phrase – which can then be modified by yet another adjective. Then you get the ambiguity about which noun the adjective modifies.
Spanish, in contrast, seems to not allow this (except that speakers of Spanglish have adopted it). Thus, where English speakers say “chicken burrito”, Spanish speakers would say “burrito de pollo” (literally, “burrito of chicken”). But Spanglish speakers would just say “pollo burrito”. Note that the Spanish grammar doesn’t give rise to this ambiguity. (It has nothing to do with the Spanish habit of putting the adjective after the noun, as I’ve occasionally heard claimed.)