Polymeric fibrous scaffolds based on the biocompatible and biodegradable three-arm-branched star poly(epsilon-caprolactone) (Mw = 189,000 g/mol) were prepared by a melt electrospinning technique. The possibility of processing polymers without the use of organic solvents is one of the main advantages over solution electrospinning. Scaffolds were biologically tested for their ability of supporting skin tissue regeneration. For this purpose, mouse embryo fibroblast (BALB/3T3 clone A31) and human keratinocyte (HaCaT) cell lines were selected as models, and seeded onto the polymeric supports both as single and co-culture. Cell viability, proliferation, and collagen production were assessed by WST-1 assay and Direct Red 80 dye, respectively. Cell morphology and colonization of the supports were evaluated by scanning electron microscopy and confocal laser scanning microscopy. Results highlighted that the star poly(epsilon-caprolactone) scaffolds were able to promote collagen production by fibroblasts. In co-culture studies, scaffolds supported adhesion, proliferation, and spatial organization of both cell lines. By virtue of the observed results, the developed polymeric scaffolds appeared suitable as biodegradable and biocompatible three-dimensional supports for skin tissue regeneration in wound healing dressing.