Kaplansky's second conjecture (on Hopf algebras) deals with "admissible" coalgebras: He calls a coalgebra admissible, if there is an algebra structure making it a Hopf algebra. The conjecture states that:

a coalgebra $C$ is admissible if and only if any finite subset of $C$ lies in a finite dimensional admissible subcoalgebra.

As far as I know there are counterexamples refuting the conjecture. (the first one was stated by Larson if I remember correctly).

However, my question has to do with whether there is some similar conjecture (or some result) for what could be called an admissible algebra : Given an algebra $A$, under which conditions can a coalgebra structure be found on $A$ such that it will be turned into a Hopf algebra? Furthermore, when would such a coalgebra structure be unique?