Why is "ain't" not listed in dictionaries?

Google finds 52,000,000 matches for ain't but non-natives simply can't look up this word. Wiktionary isn't helpful.

Is it some kind of 'wildcard' for "am/is/are not"?

The common bit of schoolyard wisdom that “ain’t ain’t in the dictionary, so ain’t ain’t a word” turns out to be untrue. Every online dictionary that I’ve ever looked in contained an entry for ain’t.

Merriam-Webster Online:

Main Entry: ain't
Pronunciation: \ˈānt\
Etymology: contraction of are not
Date: 1749
1: am not : are not : is not
2: have not : has not
3: do not : does not : did not —used in some varieties of Black English

usage Although widely disapproved as nonstandard and more common in the habitual speech of the less educated, ain't in senses 1 and 2 is flourishing in American English. It is used in both speech and writing to catch attention and to gain emphasis <the wackiness of movies, once so deliciously amusing, ain't funny anymore — Richard Schickel> <I am telling you—there ain't going to be any blackmail — R. M. Nixon>. It is used especially in journalistic prose as part of a consistently informal style <that ain't hay> <two out of three ain't bad> <if it ain't broke, don't fix it>. In fiction ain't is used for purposes of characterization; in familiar correspondence it tends to be the mark of a warm personal friendship. It is also used for metrical reasons in popular songs <Ain't She Sweet> <It Ain't Necessarily So>. Our evidence shows British use to be much the same as American.

Random House Unabridged:

ain't   [eynt] /eɪnt/
1. Nonstandard except in some dialects. am not; are not; is not.
2. Nonstandard. have not; has not; do not; does not; did not.
Origin: 1770–80; var. of amn't (contr. of am not ) by loss of m and raising with compensatory lengthening of a; cf. aren't

—Usage note As a substitute for am not, is not, and are not in declarative sentences, ain't is more common in uneducated speech than in educated, but it occurs with some frequency in the informal speech of the educated, especially in the southern and south-central states. This is especially true of the interrogative use of ain't I? as a substitute for the formal and—to some—stilted am I not? or for aren't I?, considered by some to be ungrammatical, or for the awkward—and rare in American speech— amn't I? Some speakers avoid any of the preceding forms by substituting Isn't that so (true, the case) ? Ain't occurs in humorous or set phrases: Ain't it the truth! She ain't what she used to be. It ain't funny. The word is also used for emphasis: That just ain't so! It does not appear in formal writing except for deliberate effect in such phrases or to represent speech. As a substitute for have not or has not and—occasionally in Southern speech— do not, does not, and did not, it is nonstandard except in similar humorous uses: You ain't heard nothin' yet! See also aren't.

Collins English Dictionary - Complete & Unabridged 10th Edition

ain’t (eɪnt) — contraction of am not, is not, are not, have not, or has not: I ain’t seen it

Online Etymology Dictionary:

ain't
1706, originally a contraction of am not, and in proper use with that sense until it began to be used as a generic contraction for are not, is not, etc., in early 19c. Cockney dialect of London, popularized by representations of this in Dickens, etc., which led to the word being banished from correct English.

Dictionary of American Slang and Colloquial Expressions:

ain't definition
in.
is not. (Colloquial. Stigmatized, jocular, mock undereducated, as well as undereducated use. Its use is widespread and sometimes deliberate in educated spoken use and when writing for effect. Properly an old contraction of am not or are not. The battle against ain't was lost at least two centuries ago but is still fought in isolated areas. See also If it ain't broke don't fix it.; If it ain't broke, fix it till it is.; That ain't hay!; There ain't no such thing as a free lunch.) : Saying “Ain't ain't in the dictionary” ain't so.

Cambridge Advanced Learner's Dictionary:

ain't short form /eɪnt/ not standard am not, is not, are not, has not, or have not He ain't going. "Can I have a fag?" "I ain't got none left."

ain't (ānt)

Webster's New World College Dictionary:

1. Informal am not
2. : a dialectal or nonstandard usage
   a. is not or are not
   b. has not or have not

Origin: early assimilation, with lengthened and raised vowel, of amn't, contr. of am not; later confused with a'nt (are not), i'nt (is not), ha'nt (has not, have not)

American Heritage Dictionary, 4th edition:

ain't (ānt)
Nonstandard
1. Contraction of am not.
2. Used also as a contraction for are not, is not, has not, and have not.

Usage Note: Ain't has a long history of controversy. It first appeared in 1778, evolving from an earlier an't, which arose almost a century earlier as a contraction of are not and am not. In fact, ain't arose at the tail end of an era that saw the introduction of a number of our most common contractions, including don't and won't. But while don't and won't eventually became accepted at all levels of speech and writing, ain't was to receive a barrage of criticism in the 19th century for having no set sequence of words from which it can be contracted and for being a “vulgarism,” that is, a term used by the lower classes, although an't at least had been originally used by the upper classes as well. At the same time ain't's uses were multiplying to include has not, have not, and is not, by influence of forms like ha'n't and i'n't. It may be that these extended uses helped fuel the negative reaction. Whatever the case, criticism of ain't by usage commentators and teachers has not subsided, and the use of ain't is often regarded as a sign of ignorance. • But despite all the attempts to ban it, ain't continues to enjoy extensive use in speech. Even educated and upper-class speakers see no substitute in folksy expressions such as Say it ain't so and You ain't seen nothin' yet. • The stigmatization of ain't leaves us with no happy alternative for use in first-person questions. The widely used aren't I? though illogical, was found acceptable for use in speech by a majority of the Usage Panel in an earlier survey, but in writing there is no acceptable substitute for the stilted am I not?

Longman Dictionary of Contemporary English Advanced Learner's Dictionary:

ain't a short form of 'am not', 'is not', 'are not', 'has not', or 'have not', that many people think is incorrect

Macmillan English Dictionary:

ain't spoken pronunciation /eɪnt/

a way of saying ‘am not’, ‘is not’, ‘are not’, ‘has not’, or ‘have not’. Many people consider ‘ain’t’ to be incorrect.

Other dictionaries available to me at my desk include The New Oxford American Dictionary that ships with Mac OS X, which also has an entry for ain’t:

ain't |ānt| informal
contraction of

• am not; are not; is not : if it ain't broke, don't fix it. [ORIGIN: originally representing London dialect.]
• has not; have not : they ain't got nothing to say. [ORIGIN: from dialect hain't.]
  USAGE The use of ain't was widespread in the 18th century and is still perfectly normal in many dialects and informal contexts in both North America and Britain. Today, however, it does not form part of standard English and should not be used in formal contexts.

So, clearly, non-natives (and natives too) can look up this word.

Part of the reason it doesn't make the cut is because it is considered nonstandard (why it never earns a status as a valid alternative form is tough to say). But if your dictionary is robust enough, you'll find it in there. The Oxford English Dictionary says the following:

Negative forms colloquial and vulgar, found in dramatists and novels since 17th c. are ar'n't, a'n't = are not, am not, e'n't, ain't = am not, is not, are not.

Actually, it is listed in various dictionaries. Some quotations about its usage:

Ain't is one of the most informal verb contractions in English, and its use in formal contexts may be criticized because it is associated with careless speech. It is, however, accepted in folk and popular song lyrics, show titles, direct quotations, and fictional dialogue. Otherwise ain't is best avoided, except as a deliberate rhetorical device and in allusive expressions such as You ain't seen nothing yet.

The use of ain‘t was widespread in the 18th century, typically as a contraction for am not. It is still perfectly normal in many dialects and informal speech in both Britain and North America. Today, however , it does not form part of standard English and should never be used in formal or written contexts.

Finally, the definition of ain't in Merriam-Webster Dictionary:

Etymology: contraction of are not
Date: 1749

1. am not : are not : is not
2. have not : has not
3. do not : does not : did not —used in some varieties of Black English

Basically, yes. It's almost the definitive colloquial word: every native speaker knows perfectly well what it means, but no one admits it in formal or even semi-formal usage.

It's not (as far as I know) a contraction of actual words; rather, it is a slurring of several different possibilities, which are as you listed.