The peculiar features of quantum magnetism sometimes forbid the existence of gapped "featureless" paramagnets which are fully symmetric and unfractionalized. The Lieb-Schultz-Mattis theorem is an example of such a constraint, but it is not known what the most general restriction might be. We focus on the existence of featureless paramagnets on the spin-1 square lattice and the spin-1 and spin-1/2 honeycomb lattice with spin rotation and space group symmetries in 2+1 dimensions. Although featureless paramagnet phases are not ruled out by any existing theorem, field theoretic arguments disfavor their existence. Nevertheless, by generalizing the construction of Affleck, Kennedy, Lieb, and Tasaki to a class we call "slave-spin" states, we propose featureless wave functions for these models. The featurelessness of the spin-1 slave-spin states on the square and honeycomb lattice are verified both analytically and numerically, but the status of the spin-1/2 honeycomb state remains unclear. © 2016 American Physical Society.