Field of interest: Astronomy and Astrophysics (please do share your knowledge and experiences for other fields as well)
Many journals allow the author to request for a double blind peer review.
When should one go for it? What is the current best practice?
Scientific publishing and peer review, no matter what anyone tells you, is very unfair. Therefore, double-blind vs. single-blind (or open) peer review really depends on which choice you think tip the scales in your favor (or at least tip it less in favor of rejection).
The pro of double-blind review is that nobody will judge you based on who you are, your apparent gender (based on your name), or what part of the world/university you come from. The focus will (hopefully) be primarily on the science. Thus, double-blind review seems to be the most ethical choice.
However, maybe you or your boss are big-shots, come from a major R1 university, and are generally part of the academic elite. Then you may WANT people to judge you in part based on who you are! I'm sure we all can think of a few papers in our fields authored by "the big guns" that got published in big journals despite being more or less incremental research!
Open peer review pushes this to the extreme. If you're a powerful person in a position to ruin someone's career, a hypothetical peer reviewer in an open system would be loathe to reject or harshly criticize your paper for fear of future retaliation.
As far as "best practice," I think this varies quite a bit by field. In my field (applied physics/engineering) double-blind and open peer review are only options in a couple of journals that I'm aware of.
