Where did the singular "innings" come from?

Solution 1:

"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.

Solution 2:

"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.

Solution 3:

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".