The most popular and probably most used Agile Delivery Framework is Scrum. It is that popular, that very often Scrum and Agile are confused or taken as the same thing. Whereas Agile is an overlying mindset and sense-making approach, Scrum is a framework with a set of activities, roles and outcomes (called Events, accountabilities and artifacts), that help teams to deliver value in an Agile way (albeit the process or framework itself need to have different mindset, empowerment, leadership and alignment practices being fully effective).
The key of any Scrum team to be successful is the focus on feedback, learning and improvement (or Empiricism), in which it is important, that there are measurements in place, which help them to learn. Through short feedback cycles (with sprints), they can review the delivery and the way they work and immediately suggest improvements on the delivery as well the way they work to become better in the next timeboxed cycle (Sprint). The big advantage of Scrum is to have an organized and structured approach to an Agile mindset. However, teams often forget in Scrum, is, that the end of a feedback cycle / sprint should be the latest possibility of their learning. Through daily quick interactions, teams should make quick adjustment immediately to achieve more value and better delivery.
Therefore, some teams, instead of Scrum with a fixed lifecycle, teams apply Kanban as an Agile framework. Whereas Scrum have a structure around the feedback cycles and sprints, Kanban is a framework with constant flow. While, there needs to be constant feedback to improve the delivery and way of working, often through the daily work and challenges, this gets forgotten. Scrum has a sense of celebration at the end of the sprint, in Kanban the gratification of completed work is not institutionalized. It is up to the team to evaluate and validate the delivery constantly. Whereas Scrum puts a focus on learning frequently through feedback cycles, Kanban emphasizes on learning through flow. Therefore, Kanban is only possible with a clear set of data measurement like Lead Time, Cycle Time and Limited Work in Progress. The decision of a team if they should do Kanban or Scrum depends on the nature of the work, the size as well as the maturity of the team.
Scrum or Kanban are team-based approaches and work for individual teams until a specific size (suggestions are usually up to 10 members). But, when working in bigger companies, we often need more than just 10 members working on a certain service or product. Therefore, Agile Scaling Frameworks such as SAFe, LESS, Scrum At Scale, etc. were created to give organizations options if you run more than just one team. Those Scaling Frameworks have good intentions and provide very practical tools and methods. They are usually great starting points, but every organization is different: Therefore, for organizations it is important to develop over time the own flavors of Agility in their company. Most Scaling Frameworks give unintentionally the impression, that we might achieve agility through following the frameworks. But as described before under Scrum, also the scaling frameworks require a different mindset, empowerment, leadership and alignment practices for becoming fully effective.
Therefore, organizations should focus on having teams as a whole being cross-functional, competent and self-organized instead of focusing on the process itself. Whatever frameworks is chosen, it can only be the starting point to get people used to a change in values and mindsets. The clear power of agility comes from teams being able to draw own conclusions for their context.