We will discuss the role of quenched disorder in spin-1/2 quantum magnets where magnetic frustration promotes the formation of local singlets. We find that the destruction of a valence-bond solid phase by weak quenched disorder leads inevitably to the nucleation of topological defects carrying spin-1/2 moments. This renormalizes the lattice into a strongly random network of defect spins, which yield interesting low-energy excitations. A similar conclusion is reached in a regime of stronger disorder where short-ranged valence bonds would otherwise freeze without local crystalline order. Motivated by these results, we conjecture Lieb-Schultz-Mattis-like restrictions on ground states for disordered magnets with spin-1/2 per unit cell. We apply insights from this study to propose an alternative interpretation of the phenomenology of YbMgGaO₄, which was suggested to be a quantum spin liquid. Experimental signatures, including unusual specific heat, thermal conductivity, and dynamical structure factor, and their behavior in a magnetic field, are predicted from the theory, and compare favorably with existing measurements on YbMgGaO₄ and YbZnGaO₄.