There are about 165 inherited irregular verb roots in English (for example, see, saw, seen), and maybe 35 irregular noun roots (for example, foot, feet). This does not count the Latin and Greek plurals, which we typically learn in school and don't acquire with the rest of the language.
In English, the regular nouns and verbs are the most common kind, but this isn't true in some other languages: In closely related German, for example, the overwhelming majority of nouns are irregular (the regular ending -s, although applicable to all sorts of nouns, is quite rare), and there are far more irregular verbs than in English.
Similarly, the noun classifier system in Chinese and other languages operates quite analogously to irregular noun plurals in other languages; there is a regular classifier ge, and then there are lots of fuzzily defined families of nouns, each with its specific classifier. These families tend to be organized on semantic lines, but with lots of exceptions.
For example, the Chinese classifier for human being has a respectful tone, and the word for thief doesn't normally take it, using the regular classifier instead. It's a defect in most Chinese dictionaries that they don't list the most usual classifier for a noun, in the way that French dictionaries show gender.
In Japanese, which has the same kind of classifier system, the classifier for book remains the one for vertical cylinders, despite the prevalence of codices over scrolls for some generations now. In Burmese, where nouns can almost always be used with more than one classifier, a semantic explanation tells us why basket of cows is a forbidden combination, but does not explain why a team of horses is also forbidden: one must refer to the fact that Burmese do not happen to use teams of horses.
It makes a great deal of difference whether a word is regular or irregular: compounds formed from them obey quite different rules. A compound or idiom whose head has an irregular root is irregular: overate, undid, bogeymen, stepchildren, milk teeth, straw men, oil mice, beewolves (a kind of wasp), cut a deal, bought the farm, caught cold, went bananas, threw up. Note that He threw up the ball (not an idiom) and He threw up his lunch (idiomatic) are syntactically indistinguishable; in either case, the up can be postposed.
However, a bahuvrihi compound is regular: tenderfoots, sabertooths, lowlifes, flatfoots, still lifes. Walkmans is also headless, though not technically bahuvrihi (if it were, it would mean "one who walks like a man" or something similar).
Rootless nouns and verbs made from names, quotations, sounds, abbreviations and foreign words are regular: I've been Rolling Stoned and Beatled till I'm blind, There are five "man"s on that page, The tire made several pffffts.
Denominal verbs where the verb is derived from an irregular noun are nevertheless regular: stringed means 'having had a string removed', despite the verb string, strung underlying the noun string, and to be put out (in baseball) by reason of hitting a fly ball which is caught is to be flied out, not flew out, because of the intervening noun fly 'fly ball'. Likewise, The doctor slided the sample means he put it on a microscope slide.
Regularly inflected nouns can't normally be incorporated into compounds: mice-eater is acceptable, rats-eater is unacceptable. Exceptions arise when the plural form refers to a heterogeneous collection of individuals treated as distinct entities: a dog hater needn't hate any particular dog (he need not even have met a dog), but an enemies list is a list of specific, individual enemies. Likewise with Landmarks Commission, singles bar, the Morphemes Project. Pinker collects these and puts them on his (what else?) exceptions list.