  We provide new techniques to construct sets of reals without perfect subsets
and with the Hurewicz or Menger covering properties. In particular, we show
that if the Continuum Hypothesis holds, then there are such sets which can be
mapped continuously onto the Cantor space. These results allow to separate the
properties of Menger and $\mathsf{S}_1(\Gamma,\mathrm{O})$ in the realm of sets
of reals without perfect subsets and solve a problem of Nowik and Tsaban
concerning perfectly meager subsets in the transitive sense. We present also
some other applications of the mentioned above methods.
