So I was reading Murakami Haruki's first novel Kaze no uta o kike (風の歌を聴け, "Hear the wind sing") this week, and I noticed something of linguistic interest:

「家訓なんだよ。子供はすべからく父親の靴を磨くべしってさ。」
「何故？」
「さあね。きっと靴が何かの象徴だと思ってるのさ。...」
「いい習慣ね。」
「そう思う？」
「ええ。お父さんに感謝するべきよ。」

"It's part of my upbringing. The word is children are supposed to polish their fathers' shoes."
"Why's that?"
"Beats me. Shoes must be symbolic of something. ..."
"Nice tradition."
"Think so?"
"Sure. I think it's right you should show your gratitude to your father."
(Translation: Alfred Birnbaum)

The highlighted words there, beshi and beki are actually fossilized variations on the same original morpheme, most often rendered /besi/ (although /ubesi/ might make more sense; I think I read that argument in Vovin). Back in the day, this was a derivational suffix (or "auxiliary verb" 助動詞 in the traditional classification) that, roughly speaking, added the meaning "must" or "should" or "will (surely)" to verbs:

/nomu/ → drink
/nomu besi/ → must drink/should drink/etc.

So that final /-si/ is just the standard conclusive ending for adjectives, and we should expect /besi/ to have become /bei/ in contemporary Japanese. As a matter of fact we do see that in the eastern dialects: that's what all those variants of /be/ (as in iku be, "Let's go"). But this isn't done in Standard Japanese, and instead it's fossilized into three separate forms, which derive from the old conjugations of /besi/ but aren't really perceived as such any more.

First, there's bekarazu. This is a negative form, and is used exclusively in prohibitions: hairu bekarazu, "no entry."

Second, there's beki. This is the premodern attributive form of /besi/, so originally it was used for modifying nouns (yomu beki hon, "must-read book") and not predicatively. But this distinction was lost throughout the language in the middle of the second millennium, and so /beki/_ both retained its original attributive use and became used conclusively, while the original conclusive form /besi/ vanished.

That can't be the whole story, though: if besi evolved just like other adjectives, we would expect it to have gone all the way and become /bei/ as noted above, same way that akaki ("red") became akai. Instead, beki appears to have turned into something rather irregular, sort of like onaji: can precede nouns as-is, but requires a copula at the end of the sentence.

For example, the woman in the dialogue above says beki yo, but that's because she's a woman; the standard way to say it would be beki da (yo). By comparison, you would also say onaji da, but you would never say akai da. (That's why some people object to using akai desu [for example] as a polite form for adjectives, but that's another story.)

Meanwhile, although forms like yomu beki no hon are still generally considered "incorrect," they're far from rare. It's possible that beki hasn't finished evolving yet, and will end up acting as a regular "no adjective."

The final fossilized form is beshi. That's the first one in the passage quoted above, and it is explicitly archaic. (It represents the survival of the dedicated conclusive form that vanished from living Japanese centuries ago, as noted above.) In the sentence in the passage above in which it appears, "'Kodomo wa subekaraku chichioya no kutsu o migaku beshi'tte sa", the speaker is using beshi in the sentence as one way of indicating that the "family tradition" of children polishing their father's shoes is old and formal. The language is not contemporary spoken Japanese, but rather premodern written Japanese, evoking weightiness and solemnity in a similar way to the language of the King James Bible when thrown into contemporary English. ("Thou shalt not...")

In theory language of this sort could include all the old conjugations of /besi/, but what distinguishes beshi is that it is simple enough to remember and use (mostly) appropriately as a non-inflecting unit. (Again, sort of like "Thou shalt not...")

I should probably also note that there are people who use beshi at the end of their sentences without meaning to be pompous — basically as a "conclusive" inflection of beki. But this usage is more mimicry or even pastiche than actual revival of the classical inflection system, as evidenced by the fact that you will often see it expanded to beshi da — such speakers know that the conclusive form goes at the end of the sentence, but they still treat it as a noun as noted above.

So, what was once a mostly unremarkable derivational suffix/auxiliary verb has broken down into a handful of non-inflecting descendants. In my opinion, this is probably due to a general tendency to reanalyze the morphologically unusual* /yom.ubeki/ (a la Vovin) as /yomu beki/, followed by confusion over what this newly isolated /beki/ actually is, and how to use it.