$\newcommand\R{\Bbb R}$Assuming the axiom of choice, is there an uncountable Lebesgue-measurable subset $S$ of $\R$ that contains no nonempty perfect set?

Of course, such a set $S$, if it exists, must be of measure $0$.

"The axiom of choice implies the existence of sets of reals that do not have the perfect set property", so that there is an uncountable subset of $\R$ that contains no perfect set; but can such a set be Lebesgue-measurable?