TL;DR: It was shown in this article that every hereditarily countable set has rank less than ω 2, and that if ℵ1 is singular then there are hereditably countable sets of all ranks less than 2.
Abstract: It is shown (in ZF) that every hereditarily countable set has rank less than ω2, and that if ℵ1 is singular then there are hereditarily countable sets of all ranks less than ω2.