[iconoclast]

That's a good question, on which I have little insight. The intuition is that it's probably Germanic, so I looked a few up in online German and Dutch dictionaries.

German (which I used to have an extensive knowledge of):

NAIL clipper = NAGelzange [nail (sing) pincer]
BULLfight(ing) = STIERkampf [bull (sing) fight]
COCKfight(ing) = HAHNenkampf [cocks (pl) fight]
apple TART = APFeltorte [apple (sing) tart]

Dutch (which I once had a passing knowledge of):

NAIL clipper = NAGelknipper ['nagel' looks singular]
BULLfight(ing) = STIERegevecht ['stiere' sure looks plural, but the '-e' suffix might be an obligatory juncture transition]
COCKfight(ing) = HANengevecht [definitely plural]
apple TART = APPeltaart (?) ['appel' looks singular]

First-element singularisation seems to operate with nail clipper and apple tart in both languages. Cockfighting is cock on cock, which might explain the use of plural in both languages, but bullfighting, which is human on a bull or on bulls (?), varies from one to the other. I'm afraid I don't know enough about Dutch to be totally sure that Dutch 'stiere' is plural. For example, German, sometimes uses a "transition" at the juncture of the two elements. Thus, when in my first post I gave the German word for speed limit, I accidentally omitted the '-s-' transition that is commonly added to words ending in '-keit' when they're compounded. The correct spelling is Geschwindigkeitsbegrenzung.

But we're not quite back to square one. First-element singularisation does occur in at least some cases in German and Dutch, so it does seem to be a Germanic word-formation thing. In English, it occurs in verb-object compounds, e.g. 'a CAR salesman' (but politically correct 'a car SALESperson', probably owing to the extra secondary stress in 'person', with salesman and -person themselves being compounds) is a person who sells cars. I've never seen or found an explanation in the literature that explains why. Indeed, English boasts one peculiarity that might seem to thoroughly muddy the issue. Nouns whose plural is still Germanic, i.e. "irregular", sometimes form compounds with the plural: one of my examples was 'people-hating', and not 'person-hating', but yet we say 'child-murderer' for the killer of a child or of children, and not 'children-murderer' (?) in the latter case. Can anyone else shed some light?