Protein degradation via ubiquitination is a major proteolytic mechanism in cells. Once a protein is destined for degradation, it is tagged by multiple ubiquitin (Ub) molecules. The synthesized polyubiquitin chains can be recognized by the 26S proteosome where proteins are degraded. These chains form through multiple ubiquitination cycles that are similar to multi-site phosphorylation cycles. As kinases and phosphatases, two opposing enzymes (E3 ligases and deubiquitinases DUBs) catalyze (de)ubiquitination cycles. Although multi-ubiquitination cycles are fundamental mechanisms of controlling protein concentrations within a cell, their dynamics have never been explored. Here, we fill this knowledge gap. We show that under permissive physiological conditions, the formation of polyubiquitin chain of length greater than two and subsequent degradation of the ubiquitinated protein, which is balanced by protein synthesis, can display bistable, switch-like responses. Interestingly, the occurrence of bistability becomes pronounced, as the chain grows, giving rise to all-or-none regulation at the protein levels. We give predictions of protein distributions under bistable regime awaiting experimental verification. Importantly, we show for the first time that sustained oscillations can robustly arise in the process of formation of ubiquitin chain, largely due to the degradation of the target protein. This new feature is opposite to the properties of multi-site phosphorylation cycles, which are incapable of generating oscillation if the total abundance of interconverted protein forms is conserved. We derive structural and kinetic constraints for the emergence of oscillations, indicating that a competition between different substrate forms and the E3 and DUB is critical for oscillation. Our work provides the first detailed elucidation of the dynamical features brought about by different molecular setups of the polyubiquitin chain assembly process responsible for protein degradation.