In this thesis we will prove various types of localization for some classes of one-dimensionalrandom Schrodinger operators. The central theme for all models considered will be singularity. Here, we use the term singularity mainly to refer to the possible lack of continuity in the probability distribution governing the randomness of the potential terms; although, we also deal with the other notion of singularity: that of Jacobi matrices and its counterpart, the unboundedness of the potential. In particular, we will prove spectral localization for unbounded one dimensional random Jacobi operators. Such operators are obtained by incorporating independent and identically distributed randomness into the off-diagonal terms of the standard Anderson model. The operators exhibit spectral localization if almost surely the spectrum is pure point and all of the eigenfunctions decay exponentially. We also consider so-called random word models. These generalizations of random Schrodinger operators have potential terms given by (row) vectors of bounded but random length which permits consideration of local correlations within the potential. These operators sometimes have a finite set of critical energies where the rate of localization tends to zero as one approaches these energies. Nevertheless, we will prove that these operators exhibit exponential dynamical localization in expectation on compact sets not containing any critical energies.