Monovalent impurities on graphene can be divided into ionically and covalently bond impurities. The covalent impurities with one chemically active electron cause universal midgap states as the carbon atom next to the impurity is effectively decoupled from the graphene π bands. The electronic structure of graphene suppresses migration of these impurities and makes the universal midgap very stable. This effect is strongest for neutral covalently bond impurities. The ionically bond impurities have migration barriers of typically less than 0.1 eV, which is about an order of magnitude less than their typical binding energies. An asymmetry between anions and cations regarding their adsorption sites and topology of their potential-energy landscape is predicted. In addition, the migration barrier for oxygen adatoms on graphene and their electronic structure is discussed. The barrier is found to be similar to that of monovalent covalently bond impurities.