Multifunctional fibrous scaffolds, which combine the capabilities of biomimicry to the native tissue architecture and shape memory effect (SME), are highly promising for the realization of functional tissue-engineered products with minimally invasive surgical implantation possibility. In this study, fibrous scaffolds of biodegradable poly(d,l-lactide-co-trimethylene carbonate) (denoted as PDLLA-co-TMC, or PLMC) with shape memory properties were fabricated by electrospinning. Morphology, thermal and mechanical properties as well as SME of the resultant fibrous structure were characterized using different techniques. And rat calvarial osteoblasts were cultured on the fibrous PLMC scaffolds to assess their suitability for bone tissue engineering. It is found that by varying the monomer ratio of DLLA:TMC from 5:5 to 9:1, fineness of the resultant PLMC fibers was attenuated from ca. 1500 down to 680 nm. This also allowed for readily modulating the glass transition temperature Tg (i.e., the switching temperature for actuating shape recovery) of the fibrous PLMC to fall between 19.2 and 44.2 °C, a temperature range relevant for biomedical applications in the human body. The PLMC fibers exhibited excellent shape memory properties with shape recovery ratios of Rr > 94% and shape fixity ratios of Rf > 98%, and macroscopically demonstrated a fast shape recovery (∼10 s at 39 °C) in the pre-deformed configurations. Biological assay results corroborated that the fibrous PLMC scaffolds were cytocompatible by supporting osteoblast adhesion and proliferation, and functionally promoted biomineralization-relevant alkaline phosphatase expression and mineral deposition. We envision the wide applicability of using the SME-capable biomimetic scaffolds for achieving enhanced efficacy in repairing various bone defects (e.g., as implants for healing bone screw holes or as barrier membranes for guided bone regeneration).