It is well known that classical radicals and some radical-like ideals of skew polynomial and skew Laurent polynomial rings are determined in a regular way by specified ideals of the coefficient ring. However, a detailed description of these coefficient ideals is far from easy. For example, even in the case of the ordinary polynomial ring it is not known whether the coefficient ideal corresponding to the Jacobson radical is the nil radical. In this paper the authors give many potential characterizations of the coefficient ideals for various radicals and radical-like ideals of skew polynomial and skew Laurent polynomial rings. They obtain some descriptions of these ideals in the case of the Levitzki and Wedderburn radical. They also introduce and describe some new radical-like ideals connected to T-nilpotence.