From my meager understanding of Middle Japanese, I have learned that, in essence, べき is the attributive form and べし is the predicative form (feel free to correct me if I'm wrong), but in Modern Japanese, I've definitely seen evidence that contradicts this information. Can you just use whichever one you want? What are the rules?
In modern Japanese, べし is not used except for in some fossilized expressions.
As for べき:
べき is still used, although with a slight dated feel to it. In colloquial contexts it would often be expressed as ～ないといけない instead.
べき is used as a noun:
行くべきだ [sby] should go
Or you could say that べきだ is a verbal phrase. I think it makes more sense to analyze the べき without だ as copula-dropping than an actual predicative use of べき by itself.
Note that in front of the formal noun の, the noun version is used, not the attributive version