In baseball, an inning is a team's (or both teams', depending on context) turn to bat. A game consists of 9 innings. In cricket, an innings is a team's turn to bat, a game consists of 2 or 4 innings. How did this difference in singular usage arise?
"inning" could conceivably be derived like "outing": An outing is a time when one is out, an inning is a time when one is in (to bat/to play). Where did the singular "innings" come from?
"Innings" in British usage is either singular or plural. It's just one of those words with identical singular and plural forms. It's not the only word ending with an s that's plural; consider (apart from many -ics words like physics and politics) news and (both singular and plural) means, series, species, etc. This is what Fowler's A Dictionary of Modern English Usage (British, 1926) says:
Plural anomalies. See -ICS for the question of whether words in -ics are singular or plural. Plural names of diseases, as mumps, measles, glanders, can be treated as singular or plural; chickenpox & smallpox, originally plural, are now reckoned singular. Innings, corps, & some other words in -s, are singular or plural without change in spelling, but, while corps has -s silent in singular and sounded in plural, an innings & several innings show no distinction, whence arises the colloquial double plural inningses. For the plural of Court Martial & Lord Justice, the number of porridge, & the difference between pence & pennies, see the words.
So it was special enough to invite comment (and unusual enough to invite coinages like "inningses"). The OED doesn't give any special etymology; it just comes from "in", as the "outing" you mentioned comes from "out".
In American usage, "inning" is a back-formation from the plural "innings". According to a random comment on languagehat,
… "innings". This is both the singular and plural form in cricket. It is also frequently seen this way in early baseball. For a time both "inning" and "innings" were seen used as singular, but by the 1870s or so the singular "innings" was uncommon. Nowadays it is unheard of in a baseball context.
"Inning" is actually a back-formation, caused by the mistaken belief that "innings" was a only a plural (and possibly by the general American tendency, to simplify spelling -- e.g. dropping the 'u' from "colo(u)r").
But "innings" is actually singular as well as plural, and has been used as such since at least the early 1700s.
There are very few singular, countable nouns English that end in "vowel-s" or "non-s-consonant-s". "Species" and "series" are countable nouns, but both were borrowed from Latin relatively recently, while "corps" was borrowed from French. Other than a few recent borrowed words ("lens", "corps"), "innings" is the only singular, countable English noun that I can find that ends in "non-s-consonant-s".
