In the sentence:
Shame on you, whomever is responsible for encouraging this.

Whomever is the subject of the clause describing the object of a preposition. Which case is used?
Is it whomever,or whoever?
Thanks

It's whoever. The subject of a relative clause is not required to match the case of its antecedent ("you" in this case).
It is required to be the subject of the verb "is" though. So, it is written in the subjective case (i.e. whoever).