I applied and was invited to interview afterwards. From what I have read online, the interview seems to have been of a format similar to what is standard for a trading internship at such a company.
Multiple rounds of probability questions (and some combinatorics) of increasing difficulty. One of the easier examples: what's the probability you've rolled 2 6s (after rolling 2 d6s) given that you've rolled a 6?
Interview questions [1]
Question 1
I interviewed in autumn 2021. Fairly standard probability questions. I didn't prepare, so I flunked the second round. See Mosteller's Fifty Challenging Problems in Probability with Solutions (you should be able to find a PDF online) for practice. My first-round interviewer was quite pleasant. However, my second-round interviewer arrived over five minutes late to the interviewer, sighed quite loudly when I was slower than he wanted and was overall quite rude. I was shocked by the lack of professionalism, and I even wish I had challenged him on it at the time. The interviewers can be quite poor at communicating (you can see this even in the example interview Jane Street have posted to their own website!) so make sure the question is completely clear to you before attempting to answer; don't assume anything. Lastly, if you come from a mathematical background, it'll often be tempting to try to find an exact solution. Jane Street isn't looking for this (again, see the example interview on their site). Focus on getting to a relatively quick answer that doesn't have to be too precise.
I applied online. The process took 6 days. I interviewed at Jane Street (London, England) in Sept 2022
Interview
A lot of mathematical problem solving and brain teaser type questions mainly involving everyday things such as a deck of cards or dice. You can solve using general probability and statistic concepts as well using some general knowledge on how you would expect something to turn out.