Polybutylene succinate (PBS) is a biodegradable polyester with better processability and different mechanical properties compared to polylactides (PLAs), the most commonly used synthetic polymers in tissue engineering (TE). Since only few studies have evaluated PBS-containing materials for bone TE, we prepared PLA-PBS blends and analyzed material properties as well as cell attachment, proliferation, and osteogenic differentiation of human mesenchymal stem cells (hMSCs) on scaffolds. In addition to PLA, PBS, and PLA-PBS blends, PLA-polycaprolactone and PLA-poly(trimethylene carbonate) blends were evaluated. Polymer fibers were prepared using melt spinning. Pure PBS was observed to have the highest crystallinity and strain at break compared to the tougher PLA and PLA blends. No degradation occurred during the 4-week hydrolysis in either of the materials. Knitted and rolled scaffolds were manufactured, seeded with hMSCs, and cultured for 27 days. Human MSC viability was good on all the materials, but cell spreading along the fibers was only detected in PBS-containing scaffolds. They also induced the strongest proliferative response and osteogenic differentiation, which diminished with decreasing PBS content. Based on these results, PBS is superior to PLA with respect to hMSC attachment, proliferation, and osteogenesis. This encourages utilizing PBS-based biomaterials more widely in bone TE applications.