Approximate Computing (AxC) has emerged as a promising paradigm to enhance performance and energy efficiency by allowing a controlled trade-off between accuracy and resource consumption. It is extensively adopted across various abstraction levels, from software to architecture and circuit levels, employing diverse methodologies. The primary objective of AxC is to reduce energy consumption for executing error-resilient applications, accepting controlled and inherently acceptable output quality degradation. However, harnessing AxC poses several challenges, including identifying segments within a design amenable to approximation and selecting suitable AxC techniques to fulfill accuracy and performance criteria. This survey provides a comprehensive review of recent methodologies proposed for performing Design Space Exploration (DSE) to find the most suitable AxC techniques, focusing on both hardware and software implementations. DSE is a crucial design process where system designs are modeled, evaluated, and optimized for various extra-functional system behaviors such as performance, power consumption, energy efficiency, and accuracy. A systematic literature review was conducted to identify papers that ascribe their DSE algorithms, excluding those relying on exhaustive search methods. This survey aims to detail the state-of-the-art DSE methodologies that efficiently select AxC techniques, offering insights into their applicability across different hardware platforms and use-case domains. For this purpose, papers were categorized based on the type of search algorithm used, with Machine Learning (ML) and Evolutionary Algorithms (EAs) being the predominant approaches. Further categorization is based on the target hardware, including Field-Programmable Gate Arrays (FPGAs), Application-Specific Integrated Circuits (ASICs), general-purpose Central Processing Units (CPUs), and Graphics Processing Units (GPUs). A notable observation was that most studies targeted image processing applications due to their tolerance for accuracy loss. By providing an overview of techniques and methods outlined in existing literature pertaining to the DSE of AxC designs, this survey elucidates the current trends and challenges in optimizing approximate designs.