Are gap semesters frowned upon in graduate school applications? Should I explain mental health situation that led to the gap? Disclaimer: not all of what follows is relevant to the question.

I am a math student currently doing a Master’s. I plan on taking a break in the spring semester because due to many personal circumstances, my mental health had severely declined (I attempted suicide twice), and I completely lost my focus.

I am really passionate about mathematics and I do a lot of extracurricular reading. I have a GPA of about 4 (from a fairly reputable school) and I have excellent relationships with the professors in my department, so I can guarantee strong recommendation letters (especially that they seemed totally fine with my current plan). Furthermore, although I am not from an English-speaking country, I have a very good TOEFL score, and to me, the GRE is pretty much a piece of cake.

Ultimately, my goal is to get into a top 5 school. I decided to get a Master’s for that very purpose. I am worried about two things: first, that I do not have an undergraduate research experience (I never had a real opportunity anyway, though I will have written a thesis by the end of my Master's); second, and most importantly, I am worried that this break which I am about to take will look bad on my application, especially that I will be having a lot of therapy sessions and that I have a lot of personal and social challenges to face, so I am probably not going to do any math whatsoever. Also, it doesn’t seem like I’m getting any job in the meantime because there aren’t many offers and most of my applications have been declined.   

So my question is: would such a gap semester look bad on my application? (or at least get frowned upon by the admissions committee?)

And if so: How much should I go into details about my situation? Should I mention real-life incidents, or should I just say that I had excruciating circumstances, and I had to back off and I couldn't find something interesting to do in the meanwhile?
I'm going to give a slightly different advice from the previous answers.  I don't want to counsel dishonesty here, but I think it would be very unwise to write anything in your personal statement that suggests any mental health problems, especially if the only thing that needs explaining is taking a semester off.  

I want to emphasize that I think the stigma around mental health problems is very unfair and based on ignorance.  However, the people reading these applications will have probably internalized it to some degree: if you, for example, mention attempting to commit suicide, that will be the most vivid and memorable thing from your application, whereas your master's degree ending a semester early will be barely noticeable.  I think you will probably be fine not mentioning it at all, but if you do, I would just say it is for health reasons, or to prepare for moving to another country (which are, of course, true).  One of the primary things an admissions committee will have in mind is whether a student is capable of finishing the program, while not creating a lot of trouble for people in the department. They have had pain-in-the-butt students before, and would really prefer to avoid having any more.  Mentioning "personal and social challenges" or "excruciating circumstances" (which, incidentally, sounds very strange in English; it's hard to imagine what you really mean by it) is not going to give them confidence that you won't make a lot of trouble for them.