In this paper we find conditions under which the properties of Menger, almost Menger and weakly Menger are equivalent as well as the corresponding properties of Lindelöf-type. We give counterexamples that show the interrelations between those properties. The subject of our investigation is also the preservation of almost Menger and weakly Menger properties under subspaces and products. We also consider the weaker versions of Alster space and $D$-spaces.