**Abstract**

We prove that, for any infinite-type surface S, the integral homology of the pure mapping class group PMap(S) and of the Torelli group T(S) is uncountable in every positive degree. By our results in arXiv:2211.07470 and other known computations, such a statement cannot be true for the full mapping class group Map(S) for all infinite-type surfaces S. However, we are still able to prove that the integral homology of Map(S) is uncountable in all positive degrees for many infinite-type surfaces S. The key hypothesis is, roughly, that the space of ends E of the surface S contains a limit point of topologically distinguished points. This includes all surfaces with countable end spaces that have a unique point of maximal Cantor-Bendixson rank, which is a successor ordinal. We also observe an order-10 element in the first homology of the pure mapping class group of any surface of genus 2 with non-empty boundary, answering a recent question of George Domat.