What opinions say that one should say Tikkun Chatzos? What opinions say you shouldn't?
What are the reasons behind these opinions?
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.
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.