It is well known that many, but not all, toric varieties and Deligne-Mumford stacks possess full exceptional collections of line bundles in their derived category. Line bundles with trivial cohomology appear naturally as differences in exceptional collections of line bundles on toric stacks, which naturally makes them worth studying. The very first question of whether the set of such bundles is infinite already appears to be nontrivial. While there is a simple combinatorial criterion in dimension two, the dimension three case is already not fully understood. I will state known results and open problems and indicate possible approaches. This talk is based on the papers/preprints arXiv:2312.02885, arXiv:1904.00799, arXiv:1812.01758 by multiple authors.