Computation task offloading plays a crucial role in facilitating computation-intensive applications and edge intelligence, particularly in response to the explosive growth of massive data generation. Various enabling techniques, wireless technologies and mechanisms have already been proposed for task offloading, primarily aimed at improving the quality of services (QoS) for users. While there exists an extensive body of literature on this topic, exploring computation offloading from the standpoint of task types has been relatively underrepresented. This motivates our survey, which seeks to classify the state-of-the-art (SoTA) from the task type point-of-view. To achieve this, a thorough literature review is conducted to reveal the SoTA from various aspects, including architecture, objective, offloading strategy, and task types, with the consideration of task generation. It has been observed that task types are associated with data and have an impact on the offloading process, including elements like resource allocation and task assignment. Building upon this insight, computation offloading is categorized into two groups based on task types: static task-based offloading and dynamic task-based offloading. Finally, a prospective view of the challenges and opportunities in the field of future computation offloading is presented.