The differentiation potential of mesenchymal stem cells (MSC) has been extensively tested on electrospun scaffolds. However, this potential is often assessed with lineage-specific medium, making it difficult to interpret the real contribution of the properties of the scaffold in the cell response. In this study, we analyzed the ability of different polycaprolactone/polylactic acid PCL/PLA electrospun scaffolds (pure or blended compositions, random or aligned fibers, various fiber diameters) to drive MSC towards bone or tendon lineages in the absence of specific differentiation medium. C3H10T1/2 cells (a mesenchymal stem cell model) were cultured on scaffolds for 96 h without differentiation factors. We performed a cross-analysis of the cell–scaffold interactions (spreading, organization, and specific gene expression) with mechanical (elasticity), morphological (porosity, fibers diameter and orientation) and surface (wettability) characterizations of the electrospun fibers. We concluded that (1) osteogenic differentiation can be initiated on pure PCL-based electrospun scaffolds without specific culture conditions; (2) fiber alignment modified cell organization in the short term and (3) PLA added to PCL with an increased fiber diameter encouraged the stem cells towards the tendon lineage without additional tenogenic factors. In summary, the differentiation potential of stem cells on adapted electrospun fibers could be achieved in factor-free medium, making possible future applications in clinically relevant situations.