When during the application process should a candidate mention that their spouse is also looking for a job

When during the application process should a candidate say that their spouse is also looking for a job?

The issue of academic spouses who want to find employment close enough to live together is colloquially called the "two body problem", and it is a perpetual challenge for job applicants.

The italicized question came up in a recent comment thread on this site, and I realized there is no post on Academia.SE about it. So I am making one.

I have seen many answers to the question, which usually fall between two extremes:

1. 
   

   Don't mention your spouse in any way until you have a job offer, then bring up the question of a job for your spouse. Sometimes the advice even recommends taking off wedding rings during interviews.

   
   


2. 
   

   Mention the spouse immediately in your application cover letter.

   
   

Each has its advantages and disadvantages, certainly, depending on the specific circumstances of the applicant and school being applied to.

Because many people, like me, have a perspective limited by their own life experience and the schools they have worked at, I expect that there should be many different answers to the question, expressing different perspectives. I think that collection of answers would provide an extremely useful resource for job applicants.

In the thread I linked, another user left a comment:

... I'd particularly be interested in seeing answers supported by data, as there seem to be a wide range of opinions, each supported by apparently convincing logical reasons and/or anecdotes.

I don't want to limit answers here to ones based on data, but answers that do include broad data would be particularly welcome.

This kind of question has also been asked on MathOverflow.