Imagine my surprise when, just for fun, I looked up the word, and there it was, right in my school's Thorndike-Barnhart Junior School Dictionary. It was in the dictionary. This was my introduction to the reality that there are two different editorial policies used by the editors of dictionaries. The terms we use to describe them are descriptive and prescriptive. You can probably figure out what they mean.
Descriptive dictionaries describe the language.
They include words that are commonly used even if they are nonstandard. They will often include nonstandard spellings. Prescriptive dictionaries tend to be more concerned about correct or standard English. They prescribe the proper usage and spelling of words. That school dictionary in which I found ain't was a descriptive dictionary.