(Bio)degradation in response to external stimuli (stimuli-responsive degradation, SRD) is a desired property in constructing novel nanostructured materials. For polymer-based multifunctional drug delivery applications, the degradation enables fast and controlled release of encapsulated therapeutic drugs from delivery vehicles in targeted cells. It also ensures the clearance of the empty device after drugs are delivered to the body. This review summarizes recent development of various strategies to the design and synthesis of self-assembled micellar aggregates based on novel amphiphilic block copolymers having different numbers of stimuli-responsive cleavable elements at various locations. These cleavable linkages including disulfide, acid-labile, and photo-cleavable linkages are incorporated into micelles, and then are cleaved in response to cellular triggers such as reductive reaction, light, and low acid. The well-designed SRD micelles have been explored as controlled/enhanced delivery vehicles of drugs and genes. For future design and development of effective stimuli-responsive degradable micelles toward tumor-targeting delivery applications in vivo, a high degree of control over degradation for tunable release of encapsulated anticancer drugs as well as bioconjugation for active tumor-targeting is required.