Tachanun, Ashrei, Aleinu, Shir Shel Yom
- According to Sephardim there is no Tachanun on the following days: Rosh Chodesh, the month of Nissan, Pesach Sheni (14th of Iyyar), Lag BaOmer, Rosh Chodesh Sivan until the 12th of Sivan, Tisha BeAv, 15th of Av, Erev Rosh Hashana, Erev Yom Kippur, from the 11th of Tishrei until the 2nd of Cheshvan, 8 days of Chanuka, 15th of Shevat, 14th-15th of Adar (and of Adar Sheni). 
- According to Ashkenazim, there is no Tachanun on the following days: Rosh Chodesh, the month of Nissan, Lag BaOmer, Rosh Chodesh Sivan until Issru Chag of Shavuot (i.e. the 7th of Sivan), Tisha BeAv, 15th of Av, from Yom Kippur to Issru Chag of Sukkot (24th Tishrei), Chanuka, 15th of Shevat, 14th-15th of Adar (and of Adar Sheni). Also there is no Tachanun at the mincha before these days, except that on at mincha of erev Rosh Hashana and erev Yom Kippur there is Tachanun. .
- Some have the practice not to say Tachanun on the 15th of Iyar as a safek (doubt) for the day of Pesach Sheni, however, the authorities say that this is an incorrect minhag. 
- One shouldn't recite Tachanun on Motzei Shabbat until after Chatzot.
- The following occasions also warrant no Tachanun: 1) Wedding 2) Brit Milah... 3) Siyum Maasechet
- The minhag is that women don't say Tachanun. 
- One shouldn't say Tachanun at night. If a congregation reached Tachanun of Mincha after sunset, they may say Tachanun during Bein HaShemashot, but if the congregation doesn't want to, they don't have to and one shouldn't cause a fight about this.
- According to Sephardim anytime when one doesn't say Tachanun one also doesn't say Lamnase'ach. According to Ashkenazim one doesn't say Lamnase'ach on Rosh Chodesh, Chanuka, Purim, the 14th and 15th of Adar Aleph, Erev Pesach, Erev Yom Kippur, Tisha B'Av, or in the house of a mourner. 
The Kedusha of Yotzer and Uva Letzion
- The Minhag is to be lenient to say the Kedusha of Yotzer and Uva Letzion privately without the minyan. However, it’s proper that when says by oneself one should say it with the Torah reading tune.  If there’s one is praying in a minyan that already said Kedushat Yotzer (for example one is late) one can be lenient to say it individually even without the Torah reading tune. 
- One must sit for Kedushat Yotzer and if one was previously standing one should sit. Ashkenazim hold that it’s preferable to sit but not necessary. 
- One should kiss his Tefillin Shel Yad during Kedushat Yotzer. 
- The congregation should say it out loud and if an individual says it he should say it quietly. 
- When saying Ashrei one should have the proper Kavana (concentration) especially for the pasuk Pote'ach Et Yadecha. 
- One may not leave shul before the Kedusha of Uva Letzion. 
- One should say the aramaic translation of the Kedusha of Uva Letzion quietly. 
- One should say Aleinu standing with awe and trepidation.  According to tradition, it was instituted by Yehoshua Ben Nun.
- When the congregation is saying Aleinu, one should also say it along with them. 
- One may not leave shul before Aleinu is recited at the end of davening, even if he has personally finished, unless there is a particular need. 
- Women are not obligated to recite Aleinu but it is a praiseworthy thing for them to do so.
Shir Shel Yom
- The Ashkenazic custom is to say Shir Shel Yom after Aleinu. 
↑ Yalkut Yosef 131:36
↑ see Aish.com for a list of six events which occurred on that date by Rabbi Yisrael Meir Lau
↑ Kitzur Shulchan Aruch 22:8
↑ Yalkut Yosef 131:37
↑ Kaf HaChaim 131:14
↑ Sh"t Yabia Omer 4:13 writes that one doesn't say Tachanun in the davening right before a siyum just like the mishna in Taanit (31a) says that in they made a holiday on Tu BeAv because certain families finished for the season the mitzvah of chopping wood for the Bet HaMikdash. See also Rabbi Mansour on DailyHalacha.com.
↑ Piskei Teshuvot 131:1, Tefillah KeHilchata 15:5
↑ Yalkut Yosef 131:19
↑ Ben Ish Chai (Ki Tisa n. 17)
↑ Kitzur Shulchan Aruch 25:2
↑ There’s a dispute in the Rishonim whether these kedushot are Dvar Shebekedusha that need to be said with a minyan or not. Masechet Soferim 16:12 says that these Kedushot can be said by an individual.
- ↑ Rabbi Shmuel Pinchas Gelbard in Rite and Reason: 1050 Jewish Customs and Their Sources pg. 97 brings 3 reasons for not saying tachanun after sukkot:
- Since tachanun is not recited for most of the month because of the holidays, they are not recited during the days following sukkot because it as if most of the month is comprised of holy days.
- The month is suffused with mitzvot
- Throughout history it has been a time of joy: it is a month of forgiveness, Shlomo completed the beit hamikdash etc.
↑ Pri Chadash 59:3 writes that everyone agrees that if one is praying with a minyan that already said it that one can say it individually. This quoted by Pri Megadim (A”A 59:3) Magen Giborim (Aleph HaMagen 59:3), and Mishna Brurah 59:10. However the Mamer Mordechai 59:6 argues that it seems that all the poskim (including S”A) don’t make this distinction. This quoted by Shalmei Tzibbor pg 91c, Minchat Aharon 13:5, Od Yosef Chai Shemot 7, and Kaf HaChaim 59:21. Halacha Brurah 59:10 writes that since the Minhag is to be lenient in the first place to say it privately even without Torah reading tune, in such a situation one can be lenient.
↑ Sh”t Teshuvat HaRambam (Mekisei Nirdamim, Freedman, 33, Bilav 262) writes that it’s improper to stand for Kedushat Yotzer especially when the entire congregation is sitting (he even calls it Lo Tigodedo!). The Zohar (Trumah pg 132b) writes that one should sit for Kedushat Yotzer. This is brought down by Arizal (Shaar Kavanot 2 Chazarat HaAmida 38d), Magen Avraham 59:2, Solet Belulah 59:1, Kesher Gudal 10:10, Shalmei Tzibbor 91c, Siddur 1, Chesed LeAlafim 59:2, Sharei Teshuva 59:3, Ben Ish Chai (Shemot 2). Pri Etz Chaim (Chazarat Amida 4) and Kaf HaChaim 59:20 extend this also to Uva Letzion. Thus, Halacha Brurah 59:11 rules that one must sit even if he was previously standing. However, Sh”t Rama MePano 102 writes that if one was standing and sat for Kedusha one is making a mistake and if one was sitting and stood for Kedusha it’s recognized that he’s not wise, implying that one shouldn’t change from the position he is in already. So writes Knesset HaGedolah (who limits the Rama MePano to Kedushat Yotzer), Olot Tamit 59:1, Shulchan Aruch HaRav 59:2, and Machasit Shekel 59:2. Bear Heteiv 59:3 brings down both the Magen Avraham and Rama MePano. Based on this, Mishna Brurah 59:12 writes that it’s preferable to sit for Kedushat Yotzer.
↑ Ben Ish Chai (Shemot 2) and Halacha Brurah 59:11.
↑ Ravyah Brachot 1:66, Hagahot Maimon (Tefilot Kol HaShana 3), Rama 59:3 say that Kedushat Yotzer should be said out loud. Eliyah Raba 59:4 says that this only applies with the congregation but an individual should say it quietly. Sharei Teshuva 59:3 and Mishna Brurah 59:13 quote the Eliyah Raba and Shalmei Tzibbor pg 155a who says that there’s no concern for an individual to say it aloud. Halacha Brurah 59:12 rules like the Eliyah Raba because that distinction is clear in the Ramban (Lekutim beginning of Brachot s.v. VeAni Evin).
↑ Kitzur Shulchan Aruch 25:1
↑ Kitzur Shulchan Aruch 25:4
↑ Kitzur Shulchan Aruch 25:3. Ben Ish Chai (Ki Tisa n. 16) writes that even though there's some kabbalists who say it aloud the minhag of Baghdad is to say it quietly.
↑ Ben Ish Chai (Ki Tissa, 18)
↑ Kitzur Shulchan Aruch 26:6, Chida in Machzik Bracha 132:2
↑ Teshuvat Geonim (Shaarei Teshuva 43), Chida in Machzik Bracha 132:2 and Meam Loez Yehoshua 6:15)
↑ Mishna Brurah 65:9, Yalkut Yosef (Tefillah vol 2, p. 462)
↑ Sh"t Rivivot Ephraim 1:43:1
↑ Yalkut Yosef Dinim L'isha page 38, Machaze Eliyahu 20 and Halichot Bas Yisroel 2:13
↑ Kitzur Shulchan Aruch 26:6
- However the Zohar (Yitro 34a quoted by Bet Yosef) says not to say it without a minyan. Rambam (Kriyat Shema 7:17) writes that an individual should skip it rather than say it without a minyan. However Sh”t Rambam 313 (Bilav Edition) writes that one an individual can say it. So writes Rabben Avraham ben HaRambam (beginning of Maseh Roke’ach) and Rabbenu Yerucham(3:2 pg 24d), and Mahari Avuhav (Tur 59) that the Rambam retracted from his opinion in Yad Chazaka. The following hold it’s a Dvar Shebekedusha: Rav Amram Goan hashalem 1:15 pg 33) in name of Rav Tzemach Goan by Kedushat Uva Lesion and Rav Saadiah Goan by Kedushat Yotzer, Shibolei HaLeket Brachot 13 in name of many Geonim, Piskei Riaz (Megilah 3:3), and Sh”t Rashba (1:7,5:9), while the following hold it can be said by an individual: Ravya 1:2,66, Talmedei Rabbenu Yonah 13a in name of some Geonim, Hagahot Maymonit Tefilah 7:90 in name of Rashi, Sh”t Maharam 143, Mordechai Brachot 69, Rosh (Brachot 3:18, Megilah 3:7), Tur 59, Orchot Chaim (Din Kedusha Meyushav 1), and Ritva Megilah 23b.
- Some Rishonim differentiate between Yotzer which can’t be said by an individual and Kedushat Uva LeTzion which can be said by an individual including Ran (Megilah 13b), Ramban (Lekutim Beginng of Brachot s.v. VaAni Evin S”A 59:3, Rabbenu Yerucham (3:3), and Rivavan (Brachot 45b). Bet Yosef 59:3 concludes even though the Minhag is to say it privately since there’s no Gemara on the topic one should follow the Zohar as halacha. Nonetheless, as by Vayavor (which is Dvar Shebekesha) one can say it privately with the Torah reading tune (Sh”t Rashba 1:211) so too by concerning Kedushat Yotzer one can say it privately with the Torah reading tune (Sh”t Trumat HaDeshen 8). S”A 59:3 writes that some say an individual can say Kedushat Yotzer and some say not to and one should be concerned for this opinion and say it with the Torah reading tune. The Rama adds that the minhag is to say it as an individual. Sh”t Radvaz 4:10 and Biur HaGra 59:5 rule that even saying with the Torah tune isn’t sufficient and an individual should just skip it.
- Ben Ish Chai (Od Yosef Chai Shemot 6-7) writes that the Minhag Bagdad is to say it with the Torah reading tune to satisfy the opinion of the Zohar. Kesher Gudal 10:11, Levush 59:3, Shulchan Aruch HaRav 59:2, Magen Giborim (Aleph HaMagen 59:4), Mishna Brurah 59:11, Halacha Brurah 59:10, Sh”t Yabia Omer O”C 5:7(2). The Minhag nowadays is to say it privately even without the Torah reading tune but it’s preferable to say it with the tune.