A lot depends on what you want to know about realizability. You could argue that Thom's paper settles the problem: in the mod 2 case, every homology class is realizable by a map; in the integral case, some integer multiple of every homology class is realizable by a map.
There has been recent work on trying to bound the integer we need to multiply by in the integral case. See for example the preprint https://arxiv.org/abs/2303.10240 by Vasilii Rozhdestvenskii, to appear in Journal of Topology.
For realizability by embeddings, Thom says a lot. There are some nice examples of non-realizable homology classes due to Bohr, Hanke and Kotschick in
Bohr, Christian; Hanke, Bernhard; Kotschick, Dieter, Cycles, submanifolds, and structures on normal bundles, Manuscr. Math. 108, No. 4, 483-494 (2002). ZBL1009.57043.
Note however that that paper contains a mis-statement in Remark 2 at the bottom of page 485, where it is claimed that collapse at the $E^2$ page of the Atiyah-Hirzebruch spectral sequence for oriented bordism implies that all integral homology is realizable by "immersed submanifolds". All we can really conclude is that everything is realized by a map, as in Steenrod's problem.
The situation for realizability by immersions is somewhat more subtle. András Szűcs and I gave the first examples of mod 2 homology classes not realizable by immersions in
Grant, Mark; Szűcs, András, On realizing homology classes by maps of restricted complexity, Bull. Lond. Math. Soc. 45, No. 2, 329-340 (2013). ZBL1270.57068.
We also have some results about non-realizability by maps with some presecribed set of (multi-)singularities, in the above paper and in
Grant, Mark; Szűcs, András, Homologies are infinitely complex, Topol. Methods Nonlinear Anal. 45, No. 1, 55-61 (2015). ZBL1368.57013.
Zhenhua Liu asked here on MathOverflow the very interesting (in my view) question of whether there are integral homology classes realizable by immersions but not embeddings: Integral homology classes that can be represented by immersed submanifolds but not embedded submanifolds
A lot of these questions can be approched using classical (but nevertheless difficult) obstruction theory.