In the past decade, advances in artificial intelligence have revolutionized sketch-based 3D modeling, leading to a new paradigm known as Deep Sketch-Based 3D Modeling (DS-3DM). DS-3DM offers data-driven methods that address the long-standing challenges of sketch abstraction and ambiguity. DS-3DM keeps humans at the center of the creative process by enhancing the flexibility, usability, faithfulness, and adaptability of sketch-based 3D modeling interfaces. This paper contributes a comprehensive survey of the latest DS-3DM within a novel design space: MORPHEUS. Built upon the Input-Model-Output (IMO) framework, MORPHEUS categorizes Models outputting Options of 3D Representations and Parts, derived from Human inputs (varying in quantity and modality), and Evaluated across diverse User-views and Styles. Throughout MORPHEUS we highlight limitations and identify opportunities for interdisciplinary research in Computer Vision, Computer Graphics, and Human-Computer Interaction, revealing a need for controllability and information-rich outputs. These opportunities align design processes more closely with user' intent, responding to the growing importance of user-centered approaches.