Should I do a PhD or Master in Statistics for a career in Data Science?
Interesting question, but likely to be closed as "opinion-based" or "too broad".

I personally have a math Ph.D., then started to do applied statistics in a "classical" academic environment (analyzing psychology studies), finally ended up in a Data Science-type job. I fully agree that postgraduate work in statistics would be the best preparation for a career in Data Science.

As @Vietnhi notes, there are three key ingredients for becoming a good Data Scientist:


a solid grounding in statistics
programming/hacking skills
subject matter expertise


(See also Drew Conway's Data Science Venn Diagram, which I posted and discussed here).

Of the three, it seems to me like the statistics part is hardest to learn in a non-structured environment. From what I see in my day-to-day work and on CrossValidated, people will rather easily pick up programming skills and subject matter knowledge, but understanding randomness - and this is really what statistics is all about - is very hard. (See, e.g., here.)

So as long as there are no serious Data Science study tracks available (and I agree that most offerings right now are likely bandwagons people jump on), it makes sense to concentrate on statistics (getting a structured study path via the Ph.D. program), but really work to understand the actual tools for handling large data sets, and trying to get a certain amount of subject matter expertise in at least one field. Optimally, you'd do an applied stats thesis which requires you to actually work in a specific field and apply stats there.

You may want to look at Data Science.