Negative feedback loops are ubiquitous motifs in gene regulation processes, providing cells with control mechanisms to perform key decisions. Consequently, they have received a lot of recent attention in the efficient design of synthetic gene circuits. The interconnected biochemical reactions composing negative feedback loops are discrete and random by nature, and are more accurately described by stochastic models. These models, however, often make assumptions yielding erroneous predictions, or simply miss out on important discrete stochastic effects. One such common omission is the explicit delays that separate the initiation of transcription and translation from the appearance of their corresponding functional products. Thus, delay-induced stochastic oscillations and other delay-induced stochastic discrete effects have remained relatively unexplored at the single-cell level. In our work (Zavala and Marquez-Lago, submitted), we study oscillatory and multimodal behavior in a simple gene circuit with delayed negative feedback by systematically analyzing the influence of negative feedback strength and transcriptional/translational delays on expression dynamics. We carry out single-cell simulations producing exact trajectories of the Delay Chemical Master Equation, thus avoiding any approximation errors, and demonstrate an oscillatory regime emerges through a stochastic Hopf bifurcation. Furthermore, we characterize conditions under which stochastic oscillations produce bursts, and vice versa. In conformity with previous results (Marquez-Lago et al., BiophysJ 2010) we show that the same gene circuit architecture is capable of multimodal behavior and burst-like expression. Most importantly, we describe how explicit delays produce novel non-classic effects, not yet reported in the literature.