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Shabbos Mivarchim Sivan:
Vayehi Noam: When Yom Tov falls on a weekday, including if it falls on Erev Shabbos, the entire prayer of Vayehi Noam is customarily omitted on the Motzei Shabbos directly prior to the Holiday. Thus, on the Motzei Shabbos which precedes Shavuos, Vayehi Noam is omitted.
Does one omit Av Harachamim in the event that a Bris is taking place in Shul, or if there is a Chasan in Shul?
In the Shulchan Aruch Admur rules that it is to be said even in the event of a Bris, and so rule other Poskim. However, based on the ruling of Admur in the Siddur some learn it is to be omitted in the presence of a Chasan or Bris. Others however learn that even according to the Siddur it is to be recited even in the event that a Bris is taking place in Shul or a Chasan is present in Shul.
The value of time the days preceding Shavuos:
The Rebbe notes of the Rebbe Rayatz that the period of time during the days preceding Shavuos were utterly precious to him and were hence used wisely.
In 1986 the Rebbe requested a personal request that all communities have a Farbrengen during Mincha of Shabbos Parshas Bamidbar, which was Shabbos Mivarchim Sivan. This is done due to the spirit of Matan Torah when we were one nation, united with one heart and one soul.
 Admur 284:14; Rama 284:7; Admur in Siddur
Shabbos Mevarchim Iyar: On Shabbos Mivarchim Iyar, Av Harachamim is omitted. [Implication of Admur in Siddur; See Beir Hagoleh 284 in name of Maharshal; Siddur Kol Yaakov] In the Shulchan Aruch ibid, Admur rules that it is to be recited also on Shabbos Mevarchim Rosh Chodesh Iyar.
 The reason: As terrible decrees occurred during this time period of Sefira and hence we recite this prayer. [Admur ibid; See Hisvadyos 1983 p. 1448] This refers to the decrees of the year 4856, 1096, the first Crusades. [Machatzis Hashekel 284]
 Admur 295:3
 Ata Kadosh: This omission includes also the prayer of Ata Kadosh. The reason it is accustomed to omit also Ata Kadosh is because the prayer of Ata Kadosh is said as a result of saying the prayer of Vayehi Noam. As once the Shechina resides below, as is stated in the prayer of Vayehi Noam which discusses the Mishkan, it is then fit to sanctify it with the prayer of Ata Kadosh. Thus, since Vayehi Noam is omitted, [for the reason to be explained in the next footnote] so too we omit Ata Kadosh. [ibid]
Other Opinions: The above follows the Ashkenazi custom, as brought by the Tur. However, the Sefardi custom is to only omit Vayehi Noam and Yosheiv Beseiser, and they thus begin from Orech Yamim, and then say Ata Kadosh. [Tur] Some however say the Sefaradi custom is to recite it entirely, as rule the Sefaradi sources brought in the next footnote. [Piskeiy Teshuvos 294:2]
 The reason: The reason for omitting Vayehi Noam is because within the prayer we say twice “And the work of our hands”, and hence all the days of the coming week have to be fit for work. If they are not, then we omit the prayer. [Admur ibid; See Piskeiy Teshuvos 294:2 for a thorough analyses on this subject]
Other Opinions: Some, based on Kabala of Arizal, always recite the prayer of Vayehi Noam on Motzei Shabbos, even when a Holiday falls that week, and even on Motzei Shabbos Chol Hamoed. Nevertheless, in such circumstances, they recite it quietly. Their reasoning is because Vayehi Noam affects the influence of Shabbos on the weekday which is needed every week, without exception. [Birkeiy Yosef brought in Shaareiy Teshuvah 295:2; Kaf Hachaim 295:9-10]
 Admur 284:14
 M”A 284:8
 That we omit Av Harachamim on Shabbos Mivarchim Iyar due to it being within the month of Nissan when Tachanun is omitted. [Ketzos Hashulchan ibid]
 Ketzos Hashulchan 83 footnote 13; Hiskashrus 930 footnote 9
 Kitzur Hilchos Shabbos Milluim p. 81
 Otzer Minhagei Chabad p. 290
 Hisvadyus 1986 Parshas Bechukosaiy p. 366