What opinions say that one should say Tikkun Chatzos? What opinions say you shouldn't?
What are the reasons behind these opinions?
The Kitzur Shulchan Aruch 1:5 says that everybody should say Tikun Chatzos. Those who cannot do it at midnight should endeavor to awaken before dawn to say it.
The Shulchan Aruch on 1:3 says that one who Fears heaven should get up to mourn the destruction.
The Mishna Berura (ibid) brings some (unnamed?) Achronim who say that from Shavuoth to Tish'a B'Av one need not get up early enough to say Tikun Chatzos, while others say that from Tu B'Av to Tu B'Iya one gets up. (IOW a shift of about a week from the 1st opinion.)
This, BTW, is the only hint I found that the Achronim say not to say it; and even then the dispensation is for 8 -9 weeks during the short nights.
He then mentions that those who follow the Kabala customs need to always get up on time to say Tikun Chatzos.
All the other commentators on the Shulchan Aruch also laud this practice.
The Tur on O"C 1 makes it sound like it's a nice thing to do.
אם ישכים קודם אור הבוקר לקום להתחנן לפני בוראו מה יופיו ומה טובו וטוב למי שמקדים שיכוין לשעות שמשתנות המשמרות שהן בשליש הלילה ולסוף ב' שלישי הלילה ולסוף הלילה שבאלו הזמנים הקב"ה נזכר לחורבן הבית ופיזור ישראל בין העו"ג והתפלה שיתפלל אדם באותה שעה על החורבן והפיזור רצוייה וקרובה להתקבל ויפיל תחנתו לפני המקום
ומ"ש וטוב למי שמקדים וכן' בריש ברכות (דף ג') ג' משמרות הוי הלילה כו' וכתב הרא"ש עלה וראוי לכל ירא שמים שיהא מיצר ודואג באותה שעה ולשפוך תחנונים על חורבן ב"ה כמו שנא' קומי רני בלילה לראש אשמורות.
The Bet Yosef says almost the same thing.
Since the other answers mentioned Chabad: See the Shulchan Aruch HaRav (1:2-3) who talks at great length about this custom and how proper it is to do so.
Halachapedia quotes both sides of the question:
1) In order to feel pain over the destruction of the Bet HaMikdash, every night slightly before Chatzot, one should say Tikkun Chatzot. (Shulchan Aruch 1:3) writes that it is proper for a God fearing person to be pained and agonize over the destruction of the Bet Hamikdash. Mishna Brurah 1:9 writes that the Mekubalim emphasized the great importance of waking up at chatzot to say a certain Seder of Tefillot organized by the Arizal, printed in the siddurim.
2) However, the Minhag is not to say Tikkun Chatzot and some achronim justify the minhag, nonetheless, it’s praiseworthy to say it from time to time. (Piskei Teshuvot 1:10.) Yavetz in Mor UKesiah (Siman 1) writes that the obligation only applies to Torah scholars and in Israel. The Chida in Machzik Bracha 1:3 argues that it applies to all people in all places. Kitzur S”A 1:5 writes that it’s good to say Tikkun Chatzot if a person is able to wake up at Chatzot and say Tikkun Chatzot.
Here is what seems to be the most authentic reason why chabad chassidim don't specifically say tikkum chatzos.
It is no longer accustomed amongst most Jewry to perform Tikkun Chatzos. The current Chabad custom is not to say it. Nevertheless one is to perform the service of Tikkun Chatzos during his recital of Kerias Shema Sheal Hamita. Likewise if one is awake near Chatzos he should not go to sleep and is to rather use the auspicious time to learn Chassidus, or learn a Sicha or perform an accounting of the soul.
(15) According to Sichas Kodesh 5711 p. 451 in answer to Rav Yitzchak Hutner; Rebbe Rashab in Toras Shalom p. 6 “today it is thought that Tikkun Chatzos is only to be performed by great Chassidim”. https://shulchanaruchharav.com/tikkun-chatzos/#ftnref15
I asked about this in my Chabad seminary, being careful to point out that this minhag appears in Sefer Haminhagim Chabad. The rabbi insisted that "We just don't say it; it's just not what's done," even suggesting that something might not be right about saying it. I mentioned to another learned Chabad rabbi (from an old Chabad family) that it is in Sefer Haminhagim and he was stunned. So it seems clear that @NJM's general answer (that apparently one should say tikkun chatzot, and that the minhag is not to say it) is at least correct in the particular case of Chabad.