LinkedIn Views : "How do Lean, Six Sigma, and Kaizen Compare?"

“Lean” is a methodology focused on the removal of non-value added steps from a process. To be considered value added, a process must change the form, fit or function of what is being processed, must be something the customer is willing to pay for, and must be done correctly the first time. (All rework is waste)

Six Sigma is a measure of quality of the output. As Derek mentioned, Six Sigma equates by definition to 3.4 defects per million opportunities with the focus of removing variability from the process to ensure sustainable quality. Using statistical process tools, one can see the variation in the process and address areas that require attention.

Kazien literally means “to improve”, but many people use the term interchangeably with “the continuous improvement process”. Kazien events usually run just for a few days. They are typically designed to address a specific area or challenge in an operation that needs be improved upon related to waste reduction, process improvement, quality improvements, etc.

Again, these are very brief, very high level definitions. There are volumes of books and articles written on each of them that can offer more detailed information. You can also find some excellent resources online such as the Gemba Academy, isixsigma and DMAIC tools. 
By David Abramowitz

Perhaps if we keep this simple. In my view, Lean (TPS) is a complete system founded upon people involvement and uses numerous strategies and tools to eliminate waste and shorten lead times in a sustainable manner. 6 Sigma is one such tool, used for technical problem solving by specialists whilst Lean practitioners focus more on practical problem solving by the team members. Lean Sigma, is 6 Sigma plus a few "cherry picked" tools such as elimination of waste, 5s, DMAIC, Process mapping, etc., and kaizen is a strategy by which we seek to build a mindset of always seeking to improve all parts of the enterprise.
If we think of Lean as the umbrella under which these other tools and strategies come, then we have a complete system that places a greater emphasis on people and sustainability using all of the tools and strategies, than simply "cherry picking" some of the tools and strategies, and expecting lasting change.
For me, Lean transforms the entire enterprise sustainably, whilst the others simply address some of the problems, without achieving ownership 
By Robert Downie

For relative comparisons of the three mentioned, I like to think of Lean as the elimination of "non value adding content" as defined above and Six Sigma as the improvement of "value adding content". Kaizen is commonly used to refer to a discrete short term intensive collaborative activity or event. 
By Austin Morris