Has there been a formal study about the prevalence of agile vs traditional (e.g. waterfall) software development methodology in the IT industry? And how fast the agile share is growing (assuming it is)?
Gartner might have a current report on this topic (they did one in 2008), but you have to register to do searches, and I think they charge for their reports. Here's a blog posting from 2010, mentioning their 2008 report.
I found another blog posting about agile adoption, though it is a bit old now (2008).
Here are some search results from Google Scholar on agile adoption rates.
Sorry for all the web links, but the OP wants studies, so I am pointing him at some.
The CHAOS report from Standish group is one. It is not free but if you google you can find various information about it. The surveys on this website by Scott Ambler are also very useful, e.g. 2011 IT Project Success survey