Which areas have the most competition for tenure-track positions? Being a professor in a general interest department, students from diverse backgrounds and aspirations come to me for career advice. A good number of them aspire to professors but unless they're in my own discipline I'm very hesitant to say anything. But I think it would be good to have an approximation for various fields as an advisor. For example, I know that securing a tenure-track position in CS is not as competitive as a tenure-track position in math. I assume the competition is brutal for history/literature/philosophy etc. Is there an approximate rule of thumb that I could use to advise the students?
I do not know of any datasets that would empirically answer your entire question. However, several different datasources may provide insight into your question. First, the NSF has survey and reports that may help. These provide insight into overall employment:


The NSF Conducts a post-graduate survey for Science and Engineering. Data in here may be insightful to what happens to students post graduation.  
The NSF includes funding rates in their annual report. Check out the appendices.  


Also, this Inside Higher Ed has a table of job placements by broad fields. 

I would add the following caveats to this data and other data you may use to answer your questions. In some fields most PhDs go the academy. But in others, most PhD holders go to industry or government jobs. Additionally, not all university jobs are the same (e.g., small school versus R1). I would highlight this difference to your students as well.

Finally, I would suggest checking out salaries for different departments. Many state schools make this public (or newspapers in their state though public data requests; e.g., the Texas Tribune's listing for the state of Texas). Presumably, departments that pay more have more trouble attracting and retaining faculty. 

Edit to clarify the last sentence: Engineering and physics can be closely related disciplines. However, there is a much larger job market for engineers outside of universities. Therefore, one might expect universities to pay their faculty more to account for competition from the private sector. At the University of Texas, the median salary for the Department of Physics was $105,536. The median salary for the Department of Electrical Engineering was $118,143. Following the links, there are even better break downs of the data by position (e.g., professor vs postdoc). Comparatively, the Department of English had a median salary of $84,839. 

Without speaking to the University Leadership, I can only speculate as to why some departments pay better. My guess would be the higher paying departments have more difficult recruiting and retaining because the private sector has more jobs, pays better, and often requires fewer hours.

I would also add a personal anecdote about competition for tenure track jobs. My own PhD is in Environmental Toxicology. Most of the people I knew in grad school (both at my school and others) went on to work for industry, consulting firms, or the government in roughly that order. Few went onto universities, but those that did only needed a short (1-2 year postdoc). Conversely, my friends who are ecologists often need 5-10 year postdocs to get university jobs. This is because the ecologists have much greater competition for their tenure track jobs.