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I see the ending ～まい all over the place in the JLPT books and in example phrases but I can't actually think of an example of somebody saying it or writing in an email (from SMS style messages to work messages of varying formality). (I live in Tokyo, to rule out dialect issues)
e.g.
仕方あるまい
Do people actually use the term まい in everyday speech / writing?
No, it's not really used in everyday speech. "Everyday writing" is a little ambiguous because it's mostly the form of the writing that determines the tone. To address your edit, it would be weird to use まい in a message to your friend, but I wouldn't be surprised to see it in work correspondence if only because that tends to be more formal in general. The more formal it is the less unusual it is for a term like まい to appear. Generally it's just not a spoken term and is reserved for more formal situations, however if you were to use it in conversation you wouldn't be misunderstood or anything.
In my experience though it is a term that just isn't used much at all regardless. You should be able to recognize it, of course, but I don't think I've ever seen it used outside of writing, and even then I think I usually see it as a line from a character with a particularly formal speaking style or something of that sort. As the comments note, it has a very old fashioned sort of feeling, although it does make some modern appearances in the phrase ～じゃあるまいし.
