Deep learning techniques are instrumental in developing network models that aid in the early diagnosis of life-threatening diseases. To screen and diagnose the retinal fundus and coronary blood vessel disorders, the most important step is the proper segmentation of the blood vessels.This paper aims to segment the blood vessels from both the coronary angiogram and the retinal fundus images using a single VSSC Net after performing the image-specific preprocessing. The VSSC Net uses two-vessel extraction layers with added supervision on top of the base VGG-16 network. The vessel extraction layers comprise of the vessel-specific convolutional blocks to localize the blood vessels, skip chain convolutional layers to enable rich feature propagation, and a unique feature map summation. Supervision is associated with the two-vessel extraction layers using separate loss/sigmoid function. Finally, the weighted fusion of the individual loss/sigmoid function produces the desired blood vessel probability map. It is then binary segmented and validated for performance.The VSSC Net shows improved accuracy values on the standard retinal and coronary angiogram datasets respectively. The computational time required to segment the blood vessels is 0.2 seconds using GPU. Moreover, the vessel extraction layer uses a lesser parameter count of 0.4 million parameters to accurately segment the blood vessels.The proposed VSSC Net that segments blood vessels from both the retinal fundus images and coronary angiogram can be used for the early diagnosis of vessel disorders. Moreover, it could aid the physician to analyze the blood vessel structure of images obtained from multiple imaging sources.