  We introduce new quasi-Banach modulation spaces on locally compact abelian
(LCA) groups which coincide with the classical ones in the Banach setting and
prove their main properties. Then we study Gabor frames on quasi-lattices,
significantly extending the original theory introduced by Gr\"{o}chenig and
Strohmer. These issues are the key tools in showing boundedness results for
Kohn-Nirenberg and localization operators on modulation spaces and studying
their eigenfunctions' properties. In particular, the results in the Euclidean
space are recaptured.
