Are there any formal degrees that offer training in tools for detecting plagiarism? Or on understanding fair-use, proper attribution of sources, and avoiding plagiarism?
Are there career options which focus on exposing plagiarism?
The practical aspects of detecting plagiarism in text probably fall under the purview of Computer Science as @Davidmh discussed in this answer, but developing these kinds of tools and actually applying them to detect malfeasance are very different things.
Investigative Journalism is the professional career path that pertains most directly to the issue of detecting and exposing plagiarism. Many universities offer degrees and fellowships in journalism with exactly this focus:
Dr. Stefan Weber (PhD in Communication Sciences from University of Vienna) has essentially made a career out of detecting plagiarism, mainly in doctoral and habilitation theses from important public persons in the german-speaking area. You can find some information about him online. He has written multiple well-selling books on the topic, sells plagiarism checks, and regularly appears in public discussions.
However, note that the life of somebody who frequently and very publicly raises strong allegations against professors, politicians, and other degree-holders with substantial influence is not necessarily a fun one. As you can imagine, a lot of dirt gets unearthed and thrown in his face routinely (sometimes justified, sometimes more as part of counter-campaigning by the persons he attacked).
In summary, I think he fills an important societal niche in our area, but I really wouldn't want his career.
Computer Science has the tools for it. It is finding local correspondences in a large database, allowing for some differences, and rejecting random hits and well indicated quotes. As an example of a more classical problem in Computer Science, the first part reminds me of Multiple Sequence Alignment.
Turnitin uses some Machine Learning under the hood to refine the scores; and the people working there certainly have a career in plagiarism detection.
Suggestion: If you broaden your focus to include related fields -- recognizing authorship by writing style, natural language processing to recognize similar content, recognizing significant quotes, and other approaches which might be useful for recognizing reused content -- rather than focusing specifically on plagiarism, you're MUCH more likely to find a match than if you insist on pre-selecting this one specific application of those techniques. And you're much more likely to learn about, and be involved in developing, the cutting edge of technology than if you're tightly application-focused.
Think long and hard about exactly what you want to do, exactly what you need to learn in order to do it, and how to go about learning that. If you jump too quickly to trying to implement a solution, you're likely to solve it poorly if at all.
(That's assuming you actually do want to do research in this area. If you just want to write and sell a product, that's a different topic and academia may not be the place to work on it.)
I think it depends on what you mean, want to do and why you'd be interested. Different software offer training on how to use their plagiarism detecting software. Fair use issues are tackled in library training and in education programs. The Illinois online network/U of Illinois Springfield offer a course on copyright issues for online educators- http://www.ion.uillinois.edu/courses/catalog/ . If you are interested in just understanding those issues, then look for courses like that and consider education or library science depending on what you'd want to do. If there is a specific sphere like online ed, you could look into those types of programs.
If you'd want to teach others about those things, then look at courses like this, library workshops and staff positions in your spheres of interest and check the backgrounds and education of those currently employed in those positions.You could also look at academic support centers & centers for teaching and see who is giving workshops or services related to your area or interest. We often talk about plagiarism and fair use in the sphere of education, which is why looking into a degree in one of many education specializations might give you some of this training. However, it will depend on what you really want to do.
For instance, you could also look into copyright law and fair use policies and then go into the policy side with a law or ed policy or higher ed degree. You could also look into a rhetoric, English or professional comm program.
Though the OP doesn't seem to mention it, if you'd be interested in researching detection software and possible developing, or testing it, then you could look at a different set of degree programs. For testing such software and looking at how users interact with it you could look at human computer interaction programs. Such programs may allow a degree of development and design as well.
Or to develop programs you could go in a number of directions. If you are interested from a purely tech side, you could try computer science and related disciplines. However, something like computational linguistics with a focus on natural language processing or some other tech/linguistics combination would give you the theory, application and tech components to create, test, evaluate or implement a plagiarism software from a variety of angles. If that is something you'd be interested in then perhaps try something like applied linguistics and technology at Iowa State - http://www.engl.iastate.edu/teslapplied-linguistics/ or similar programs. You could also broaden the idea of detecting plagiarism to detecting linguistic fingerprints and look into forensic linguistics.
potential areas: forensic linguistics, applied linguistics, computational linguistics, english, rhetoric, professional communication, computer science, human computer interaction, higher ed, education, educational policy, law....
What you do with the degrees will vary widely depending on the direction you go in. It could be your own company, an admin uni position, teaching, support staff, library,developer, working for a tech company, an ed company,...