We derive systematically the canonical forms of the Quispel-Roberts-Thompson (QRT) mappings and obtain two new forms which were absent in the previous classifications. They correspond to what, in QRT parlance are called, asymmetric mappings. We derive the generic discrete Painlev{\'{e}}equations associated with these canonical forms, as well as some non-generic ones. We study the singularity structure of one of the latter and show that it possesses anticonfined singularities, a new phenomenon for discrete Painlev{\'{e}}equations.