This is the approved revision of this page, as well as being the most recent.
The Rabbis instituted certain practices to remind oneself of the destruction of the temples.
Tearing Your Clothing
Seeing the Kotel nowadays
- One who sees the Kotel in it’s destruction should rip his shirt a tefach and not have it sown up. Many poskim say that this still applies nowadays.  There is a dispute as to what a person has to see to be obligated in tearing. 
Old City of Jerusalem
- Some poskim say that one should tear upon seeing Jerusalem nowadays, while some say one doesn't need to. 
The Cities of Judah
- The custom nowadays is not to tear one's clothing upon seeing the cities of Judah, though some do on cities not under Jewish sovereignty. 
Who is obligated?
- A blind person isn't obligated to tear his clothing in mourning upon seeing on of these sights. 
- Children do not tear their clothing upon seeing these sights, even if they have already reached the age of chinuch
- One is obligated to tear his garments if he visits on a Friday. 
- The practice nowadays is not to tear during Chol Hamoed. 
Leaving one's home unfinished
- The Rabbis established that after the destruction of the temple it is forbidden to build a house that's completely plastered like palaces of kings rather one should coat it with lime and plaster and leave an Amah by Amah area unfinished adjacent to the doorway.  However, the Minhag is to be lenient and permit any type of structure as long as there is an Amah by Amah left unfinished. 
- Many in Klal Yisrael are unaware of this halacha and have their house completely finished. Some lament the Minhag and some defend the Minhag. 
- The blank portion should preferably be at least one amah by one amah. However, some poskim write that an area of a square amah is sufficient even if one side is less than an amah [e.g. 2 x ½ amot, which is an area of a square amah] 
- If the house is bought one doesn't need to change the structure to leave an Amah by Amah unfinished. 
Leaving something out at a meal
- Some dish should be lacking at every meal (even one of guests) in memory of the Churban, this dish that one intentionally leaves out can be a small or not significant dish. 
- Some hold that one shouldn't use one's best dishes execpt for on Pesach as a Zecher LeChurban 
- Ten Minute Halacha on Tearing Keriya at the Kotel by Rabbi Aryeh Lebowitz
Listening to Music
- See the separate article at this link Listening to Music
- S”A O”C 561:2 writes that when one sees the place where the Mikdash used to stand and now remains in it’s destruction, one must rip one’s shirt a Tefach. Even though Sh”t Bet Mordechai 33 is lenient not to rip one’s shirt nowadays since Har HaBayit is under Israeli control as is Rav Shlomo Aviner (Tal Chermon on Moadim, page 218) almost all achronim including Rav Ovadyah in Chazon Ovadyah (Arba Taniot pg 438, Aveilut v. 1 p. 266), Rav Moshe Feinstein in Sh”t Igrot Moshe O”C 4:70:11, Rav Vosner in Sh”t Shevet HaLevi 7:78, Rav Hadayah in Sh”t Yaskil Avdi 8:25:4(4), and Zichron Betzalel pg 143 agree that even though Har HaBayit is in Israeli control, it’s still an obligation to rip one’s shirt upon seeing the Makom HaMikdash. Moadim UZmanim (Vol 7 pg 209-11) writes that the Minhag not to rip one’s shirt can rely on those who say that really the ground that the Bet Hamikdash once stood and not the Kotel which is just a wall of Har HaBayit. (so writes Rav Tukachinsky in Eretz Yisrael 22:5). Another leniency which isn’t LeHalacha but just to defend the minhag is nowadays we don’t feel the churban like those who live in Yerushalyim unless it’s our first time or it’s been 12 months. Sh”t Mishnat Halachot 6:110 defends the minhag similarly by saying that in our sorry state a person would feel worse for the loss of his shirt than for the loss of the churban, however, someone who feels for the loss of the Churban should rip the shirt and fulfill the Halacha.
- R. Yisrael ben Shmuel of Shklov in Pe'at HaShulchan 3:2 maintains that one should tear one's garments upon seeing the Dome of the Rock.
- Shulchan Aruch 561:1 based on the gemara in Moed Katan 26a says that if one sees Jerusalem in its destruction he should tear his shirt, and say ציון מדבר היתה ירושלים שממה. Rav Hershel Schachter Bi'ikvei Hatzon 18 though, says that the custom nowadays is to be lenient with this. Rav Moshe Feinstein Iggerot Moshe 4:70:11 also says it is only considered in its destruction if arabs rule over it. Rav Shlomo Zalman Auerbach though, rules that since Jerusalem has become an international city and the Jewish government doesn't have the power to fully govern the non-Jewish worship houses in Jerusalem it is still considered in its destruction and one should tear his shirt.
- Rav Hershel Schachter Bi'ikvei Hatzon page 105 based on the Bach 561 who says that we tear because sovereignty over the cities of Judah represents our sovereignty over all of Israel and therefore since we control Israel now, there is no need to tear for Judah. This is also the ruling of Rav Shlomo Yosef Zevin in Moadim Bihalacha page 371. Although Shulchan Aruch 561:1 based on the gemara moed katan 26a says that if you see the cities of judah in their destruction you should tear your clothing and say ערי קדשך היו מדבר , in the Beit Yosef OC 561 he brings up the possibility that one doesn't need to tear his clothing even if it is inhabited by non-Jews but concludes that really one should if it is inhabited by non-Jews. Magen Avraham 561:1 and Mishna Brurah 561:2 writes that even if it is inhabited by Jews but they are ruled by non-jews one should tear. Which implies that in a situation like today's, where the government and inhabitants are Jewish one wouldn't tear.
Regarding Chevron, Rav Shlomo Aviner believes that the city is not under Arab sovereignty, but rather under autonomous sovereignty, which itself is reason to tear Keria and say Baruch Dayan HaEmmet. HaRav Dov Lior holds that both Chevron and Beit Lechem are under Arab sovereignty and one must, therefore tear Keria on them. See further Gray Matter II pg 67 and a related article by Rav Chaim Jachter.
- Tzitz Eliezer 16:39, Rav Yechiel Michel Tucatzinski in Ir Hakodesh Vihamikdash 17:1
- Rav Hershel Schachter - Are Children obligated in observances that commemorate the destruction of the beis hamikdash based on Magen Avraham 561:6. see there for an interesting explanation why chinuch would not apply here
- Rav Schachter Bi'ikvei Hatzon 18 says that some people think that they wouldn't be obligated because the Rama YD 399:3 says that one doesn't mourn on erev pesach after mid-day. However, the Or Zarua Hilchot Aveilut 432 says this is only because of the korban pesach so this halacha wouldn't apply to a regular erev shabbat or erev yom tov.
- Rav Yechiel Michel Tukatzinski Ir HaKodesh V'HaMikdash 17:1
- Gemara Bava Batra 60b details the rabbinic institutions that Chazal made as a commemoration of the Churban. One of the laws was that it was forbidden to build a sturdy house like a palace. Another institution was to leave an Amah By Amah of the house unfinished. Such is the opinion of the Rambam (Hilchot Tanit) and Shulchan Aruch 560:1.
- Darkei Moshe says that the Minhag is to be lenient in this regard like the opinion of the Tur. Magen Avraham 560:1, Mishna Brurah 560:1 and Kaf HaChaim 560:3 concur.
Kovetz Ohr Yisroel 4, p. 89 writes that you can also leave an amah by an amah unpainted to fulfill this institution
- Shaarei Teshuva laments the fact that so many are unaware of this law and don't abide by it. Aruch HaShulchan defends the lenient practice considering the fact that the first Beraita in the gemara holds that if the building is made out of a mixture of sand in the lime/cement there is no requirement to leave an Amah by Amah unfinished. Kaf HaChaim 560:11 concurs. Aruch HaShulchan argues that the second Braitta which is quoted as halacha by the Rif, Rosh, Rambam, and Shulchan Aruch agrees with that idea that for a mixture it's permissible. The Aruch HaShulchan brings support from the Nemukei Yosef and Ran and leaves it as a Tzarich Iyun why Shulchan Aruch didn't mention this leniency.
- Rav Yisroel Belsky (Piskei Halachos, Vol 1, p. 113, quoting Sefer Habayis, p. 215, footnote 11)
- Shulchan Aruch 560:1
- Shulchan Aruch 560:2 rules that the meal even if there are guests should be lacking a little bit. Mishna Brurah 560:6 says that it's sufficient to leave out a small dish such as Kasa DeHarsana, some type of fish which is a minimum for the requirement of Oneg Shabbat (Mishna Brurah 242:2).
- Mishna Brurah 472:5 says that it's preferable to use dishes that aren't one's best as a Zecher Lechurban except for on Pesach.