According1 to the custom of Nussach Sephard, whenever Tachanun and Nefilas Apayim are omitted one does not say למנצח...יענך (p. 71) nor תפלה לדוד הטה (p. 75). This applies on the following days: The whole month of Nissan; Pesach Sheni (the 14th of Iyar); Lag BaOmer; from Rosh Chodesh Sivan through the 12th of Sivan, i.e., five days after the festival of Shavuos, because in Temple times the Shavuos sacrifices could still be offered throughout that time, as if this were a seven-day festival;2 the Fifteenth of Av; erev Rosh HaShanah; from erev Yom Kippur until the end of the month of Tishrei; the Fifteenth of Shvat; on any day on which there is a circumcision in the synagogue, or when a bridegroom during the seven days of his wedding celebration is present; and of course on Tishah BeAv, Purim Katan (the 14th and 15th of Adar I), Purim, Chanukah and Rosh Chodesh. Tachanun and Nefilas Apayim are also omitted at Minchah on erev Rosh Chodesh, on erev Chanukah, on the eve of both Purim Katan and Purim, on erev Lag BaOmer, on the eve of the Fifteenth of Av and on the eve of the Fifteenth of Shvat. However, they are said at Minchah on the day before erev Rosh HaShanah and on the day before erev Yom Kippur, and on erev Pesach Sheni.

Tachanun is likewise omitted on the following days (in their calendar order): the Tenth of Kislev3 and the preceding Minchah; the 19th and 20th of Kislev4 and the preceding Minchah; and the 12th and 13th of Tammuz5 and the preceding Minchah.

On the yahrzeit of any of the Rebbeim one says Tachanun, unless there is an unrelated reason for not doing so.6

After Shir shel Yom, the daily Psalm, one says Hoshieinu (p. 76) throughout the year — on weekdays, Shabbos, festivals, Rosh HaShanah and Yom Kippur.7

Even if no one in the synagogue is specifically obliged to say Kaddish, each Kaddish is to be said at its respective stage in the service where our custom so prescribes. This includes the Kaddish that follows Shir shel Yom, Kaveh and Aleinu, and, moreover, even the Kaddish that follows the reading of the daily allotment of Tehillim which has been explicitly instituted as a daily obligation after Shacharis.8

“When any one of the Rebbeim used to say Kaddish, others who were also obligated to say Kaddish would not do so until the Rebbe had concluded. It is thus clear that such a pause is not considered an undesirable interruption.”9

On Rosh Chodesh, we say ברכי נפשי (Tehillim 104; p. 79) after the Song of the Day.10

From Rosh Chodesh Elul through Hoshana Rabbah one says לדוד ה' אורי (p. 81) at Shacharis after the Song of the Day, and at Minchah before Aleinu.133 This takes effect from the first day of Rosh Chodesh.11

According to our custom, Kaddish is said only after the completion of either or both of the above Psalms that follow directly after the Shir shel Yom.12

At the conclusion of each prayer service, and likewise during the Mussaf of Rosh HaShanah [and Yom Kippur], it is our custom in Aleinu (p. 84) to say שהם משתחוים להבל ולריק, but we omit the phrase that begins, ומתפללים. The reason for which one discharges saliva after the former phrase is that through speech saliva accumulates, and in this case one does not wish to benefit from it.13

After Aleinu and Al Tira (p. 85) we add the verse that begins, אך צדיקים.

On those days when Tachanun is not said, and therefore the Psalm beginning למנצח (p. 71) is also omitted, it is recited before the daily reading of Tehillim — not as part of the formal prayer service, but as an independent supplication.14

The custom which the Alter Rebbe received from his Rebbe, who received it in turn from the Baal Shem Tov, is well known — that after Shacharis and before the daily reading of Tehillim, one recites the Psalm that corresponds to one’s age. [E.g., one who is 49 years old recites Psalm 50.]15

It is the custom of certain chassidim [in practice: of chassidim at large] to read in addition the Psalm that corresponds to the age of the Rebbe.16

The morning study of Mishnayos by a mourner during the first eleven months, and likewise by a person who is observing yahrzeit, takes place after the reading of Tehillim that follows Shacharis.17 Ch. 24 of Tractate Keilim (p. 413) and ch. 7 of Tractate Mikvaos (p. 416) are studied after Maariv, Shacharis and Minchah.18

During a period of mourning, only one mishnah is studied audibly. One makes a point of completing the above chapters after the service,19 [the bulk of the text having been studied privately beforehand].

One concludes this study by quoting the mishnah that begins, ר' חנניא בן עקשיא...ויאדיר, after which one adds in a whisper a few lines of Tanya before saying Kaddish DeRabbanan.20

In the house of a mourner one recites Psalm 49 after Shacharis and Minchah.21