We consider a new type of oscillations of discontinuous unpredictable solutions for linear impulsive nonhomogeneous systems. The models under investigation are with unpredictable perturbations. The definition of a piecewise continuous unpredictable function is provided. The moments of impulses constitute a newly determined unpredictable discrete set. Theoretical results on the existence, uniqueness, and stability of discontinuous unpredictable solutions for linear impulsive differential equations are provided. We benefit from the B-topology in the space of discontinuous functions on the purpose of proving the presence of unpredictable solutions. For constructive definitions of unpredictable components in examples, randomly determined unpredictable sequences are newly utilized. Namely, the construction of a discontinuous unpredictable function is based on an unpredictable sequence determined by a discrete random process, and the set of discontinuity moments is realized by the logistic map. Examples with numerical simulations are presented to illustrate the theoretical results.