First-countable Lindelöf scattered spaces

We study the class of first-countable Lindel\"of scattered spaces, or "FLS" spaces. While every $T_3$ FLS space is homeomorphic to a subspace $\mathbb Q$, $T_2$ spaces turns out be surprisingly rich. Our investigation these reveals close ties $Q$-sets, Lusin sets, and their relatives, cardinals $\mathfrak{b}$ $\mathfrak{d}$. Many natural questions about turn independent $\mathsf{ZFC}$. prove that there exist uncountable with height $\omega$. On other hand, an finite exists if only $\mathfrak{b} = \aleph_1$. some independence results concerning possible cardinalities what ordinals can space. Several open problems are included.