Motivic cohomology. Park in Regulators article says that one hopes for an AHSS for any Noetherian scheme, and we might ask for the right notion of motivic cohomology for example in the non-reduced case, in the singular case, and in the case of an arithmetic base.
Try to explain the historical development of motivic cohomology and the challenges of finding a good notion in cases such as non-reduced, singular, or arithmetic base.
Motivic cohomology: Motivic cohomology, Voevodsky motives, Borel-Moore motivic homology, Motivic cohomology with compact supports, Higher Chow groups, Etale motivic cohomology, Motivic homology, Singular homology of varieties, Borel-Moore homology, Homology of schemes, Suslin homology, Chow groups, Friedlander-Suslin cohomology, Chow groups with coefficients, Rost's cycle modules
People search for a good notion of motivic cohomology for more general geometric objects. For example Spitzweck’s construction for arithmetic schemes, Additive Chow groups and Higher additive Chow groups for non-reduced schemes, and some attempts for stacks I think.
nLab page on D70 Motivic cohomology and related theories