Is the powerset of a regular set also a regular set?
No.  In computer science, a language is normally defined to be a subset of $\{0,1\}^*$.  If $L$ is a language, then the powerset $2^L$ is not a subset of $\{0,1\}^*$, so it is not a language.

(See e.g., https://en.wikipedia.org/wiki/Formal_language#Definition.)

If $L$ is finite, then the powerset $2^L$ is finite, so as Emil Jeřábek points out, there is a way you can encode it so its encoding is a regular language, since all finite languages are regular.

If $L$ is infinite, then as Yuval Filmus points out, the powerset $2^L$ is not countable, so there is no way to encode it as a language.  Thus, when $L$ is an infinite regular language, then the answer to your question is "no, not even if you're allowed to choose a reasonable encoding".