For a topological property \(\mathcal P\), a space \(X\) is star \(\mathcal P\) if for every open cover \(\mathcal U\) of \(X\) there exists \(Y \subset X\) such that \(X=\text{St}(Y,\mathcal U) = \bigcup\{U\in\mathcal U: U\cap Y\neq\emptyset\}\), and \(Y\) has property \(\mathcal P\). Continuing the research done by O. T. Alas, L. R. Junqueira and R. G. Wilson in [Topology Appl. 158, No. 4, 620–626 (2011; Zbl 1226.54023)], the authors study star countable and star Lindelöf spaces establishing, among other things, that there exist first countable pseudocompact spaces which are not star Lindelöf. Also, some classes of spaces are described where star countability is equivalent to countable extent, and it is shown that a star countable space with a dense \(\sigma \)-compact subspace can have arbitrary extent. It is proved that for any \(\omega _{1}\)-monolithic compact space \(X\), if \(C _{p }(X)\) is star countable, then it is Lindelöf. The paper concludes with a list of related open problems.