Thanks to advancements in deep learning, speech generation systems now power a variety of real-world applications, such as text-to-speech for individuals with speech disorders, voice chatbots in call centers, cross-linguistic speech translation, etc. While these systems can autonomously generate human-like speech and replicate specific voices, they also pose risks when misused for malicious purposes. This motivates the research community to develop models for detecting synthesized speech (e.g., fake speech) generated by deep-learning-based models, referred to as the Deepfake Speech Detection task. As the Deepfake Speech Detection task has emerged in recent years, there are not many survey papers proposed for this task. Additionally, existing surveys for the Deepfake Speech Detection task tend to summarize techniques used to construct a Deepfake Speech Detection system rather than providing a thorough analysis. This gap motivated us to conduct a comprehensive survey, providing a critical analysis of the challenges and developments in Deepfake Speech Detection. Our survey is innovatively structured, offering an in-depth analysis of current challenge competitions, public datasets, and the deep-learning techniques that provide enhanced solutions to address existing challenges in the field. From our analysis, we propose hypotheses on leveraging and combining specific deep learning techniques to improve the effectiveness of Deepfake Speech Detection systems. Beyond conducting a survey, we perform extensive experiments to validate these hypotheses and propose a highly competitive model for the task of Deepfake Speech Detection. Given the analysis and the experimental results, we finally indicate potential and promising research directions for the Deepfake Speech Detection task.