The answer to this question lies in the cultural make up of the receiving organisation and the nature of the project(s) being delivered. There is no 'one size fits all' approach.
Where unique cultural and project characteristics are ignored, the application of one methodology over another by project ‘purists’ can cause real issues with delivery, particularly where a methodology is forced upon unaccustomed participants with insufficient time and coaching to adjust (think square peg / round hole).
Consider where a software development team operates as ‘Agile’ within an organisation, but the Marketing group who support them or the Finance group who fund them, does not. Or think about how a software development project benefits from Agile because of the fluid nature of the requirements set but how, comparatively, COTS application implementations can benefit from the more rigid structure of a Waterfall delivery due to early fixed requirements and the advantages that come from longer term detailed planning.
Which methodology best suits these scenarios?
The answer is that common sense should prevail. Instead of applying a 'one size fits all' methodology, effective Project Leads will tailor a delivery strategy that aligns cultural parameters with the specific needs of each project.