The Chevra Kadisha fast:[1]

It is customary for the members[2] of the Chevra Kadisha [both male and female[3]] in each city to choose a day throughout the year to fast and discuss matters relating to burial. [Regarding if the fast falls on Erev Shabbos, see our corresponding Sefer “The Laws and Customs of Erev Shabbos.”]

When: The custom of many cities is to fast on the 15th of Kisleiv, and so is the Chabad custom.[4] The custom of Jerusalem, and other cities in Israel, is to fast on the 7th of Adar, which is the Yartzite of Moshe Rabbeinu.[5] [In a leap year, it is done on the 7th of Adar Sheiyni.[6] However, some communities fast in Adar Rishon[7], which is the date that Moshe’s Yartzite is commemorated during a leap year.[8]]

Accepting the fast by the Mincha before:[9] The fast is to be accepted from the Mincha of the day before.

The Davening:[10] They recite Selichos, Aneinu in Mincha, and Kerias Hatorah in the Davening during this day, if they are a Minyan of people fasting.

Topics of discussion: They are to use the day to discuss the Jewish burial laws that require fixing or enhancement, as well as monetary matters relating to their activities and burial plots.

Kivrei Tzadikim: They visit Kivrei Tzadikim on this day and ask for forgiveness from the deceased if they did not act with them appropriately.

Seudas Mitzvah:[11] After the fast, it is customary to make a large feast for the members of the Chevra Kadisha.

Fast falls on Erev Shabbos: See https://shulchanaruchharav.com/halacha/fasting-on-erev-shabbos/ 

Saying Aneinu:

https://shulchanaruchharav.com/halacha/is-aneinu-be-recited-in-shemoneh-esrei-of-a-private-fast-day/

[1] See Nitei Gavriel Vol. 2 chapter 97-98

[2] See Yoman of Rav Chadakav that due to that he was a member of the Chevra Kadisha, he once asked the “Ramash” if he is required to fast on the 15^th of Kisleiv, especially when it falls on Erev Shabbos, and the Rebbe replied that being that he is only a member of the Chevra Kadisha and is not one of the Gabaim, and hence he has no dealings with the actual burial of the dead, therefore he is not required to fast. The Rebbe  then said that he too does not fast, although since he fasted in the past therefore he should do Hataras Nedarim. Seemingly, the only thing they can be learned from this response is that a person who is not actual dealings with the dead, in terms of the burial and Taharah and is only a member on paper, therefore he is not required to fast. However, anyone who deals with the dead, including if they only to the Taharah, then they are to fast

[3] Nitei Gavriel 97:4

[4] Darkei Chesed p. 285; Nitei Gavriel ibid that so was custom of Liozna, Kapust, Russia, Halbershtat, Volozhin, Lita, Shklov, New York,

[5] See Nitei Gavriel Vol. 2 chapter 97 in length

[6] Nitei Gavriel 97:2 and footnote 4 in name of Sefarim

[7] See Nitei Gavriel ibid footnote 4

[8] Magen Avraham 580:8; Teshuvos Mahril 31; Terumos Hadeshen 244; Rama 568:7; Aruch Hashulchan 580:3; Mishneh Berurah 580:15; Orchos Chaim; Likkutei Sichos 16 p. 344 footnote 39: This is the final ruling, as the majority of Jewry lived in their provinces and testified this to be the custom.

Other Opinions: Some Poskim hold that during a leap year Moshe’s Yartzite is to be commemorated on Zayin Adar II. [Chacham Tzevi, brought in Sheilas Yaavetz 1:117; Siddur Yaavetz; Chasam Sofer 163; Vayaan Yitzchak 39].

[9] See Michaber and Rama 562:5; Nitei Gavriel 97:5

[10] See Darkei Chesed p. 285 for the Nussach of Selichos said on this day, and the prayers said upon visiting the cemetery; See Nitei Gavriel 97:3 and 7; Gesher Hachaim p. 352

[11] Nitei Gavriel 97:11