When to recite Kiddush Levana in months that Motzei Shabbos is past the 10th from the Molad:
One is to recite Kiddush Levana only on Motzei Shabbos [or Motzei Yom Tov]. [The above time is only with regards to Mitzvah Min Hamuvchar, however from the letter of the law it may be said on any night of the week. Certainly if one did not recite it on Motzei Shabbos, he must recite it during the week.]
The law if Motzei Shabbos falls past the 10th of the month:
One is to only delay saying Kiddush Levana until Motzei Shabbos if Motzei Shabbos falls prior to the 11th of the month. If, however, Motzei Shabbos falls on the 11th of the month or onwards, then it should be said beforehand [by a weeknight before the 11th, in nice clothing, even if he will have to say it without a Minyan]. If it falls on the 10th of the month, one is to delay Kiddush Levana until Motzei Shabbos which is the 10th. However, in a case that the Molad fell out prior to the first day of the month, in a way that the 10th of the month is the 4th night left to say Kiddush Levana [see previous Halacha for end time of Kiddush Levana], then one is not to delay saying Kiddush Levana until Motzei Shabbos if Motzei Shabbos is the 10th of the month.
Whenever Saturday night falls past the 10th day from the start of the Molad, and certainly if it falls past the 10th of the month, then one is to say the blessing of the new moon during the week and not wait until Saturday night to say.
The practical application:
Accordingly, in months that the first Motzei Shabbos [past the 7th from the Molad] will fall past the 10th of the month or Molad, then the Gabaim and Rabbanim of the Shul are to inform and arrange with the community to recite Kiddush Levana during one of the weeknights which is past the seventh of the month, or even on the previous Motzei Shabbos which is past the third day from the Molad if it is during the rainy season and the skies are often cloudy.
If reciting Kiddush Levana on a weeknight prior to Motzei Shabbos which falls past the 10th of the month will cause one to say it without Minyan, should he nevertheless say it alone during the week, or in such a case should he wait until Motzei Shabbos even though it is past the 10th of the month, in order to say it with a Minyan?
He is to say it during the week without a Minyan even in such a case.
 Michaber 426:2; Tur; Miseches Sofrim
Other Poskim: Some Poskim rule one is not to delay saying Kiddush Levana until Motzei Shabbos, as one is not to delay fulfilling a Mitzvah simply in order to fulfill it in a more enhanced way. [Bach; Chayeh Adam 118:14; Maaseh Rav of Gra; Kitzur SHU”A 97:9; brought in Kaf Hachaim 426:23 and Biur Halacha “Motzei Shabbos”; See M”A 25:2; M”B 426:20; See also Admur 25:4 that we do not delay a Mitzvah simply to do it in a greater fashion.]
 P”M 426 A”A 2; M”B 426:5 in name of Elya Raba; and so is implied from Teshuvos Harameh 78. [Kaf Hachaim 426:22]
 The reason: As at that time one appears elegant and is wearing nice clothing. [Michaber ibid] Alternatively, some say the reason for saying Kiddush Levana specifically on Motzei Shabbos is because on Motzei Shabbos the Beis Hamikdash was destroyed and the Shechina exiled. Therefore, we say Kiddush Levana on Motzei Shabbos which proclaims the return of the Shechina. [Peri Eitz Chaim 19:3; Kaf Hachaim 426:21 in name of Rav Chaim Vital] Alternatively, the reason is because the moon was diminished on Motzei Shabbos. [Tzafnas Paneiach on Rambam Barchos 10]] Alternatively, the reason is because on Motzei Shabbos there are many people found in Shul.
 M”B 426:4; Thus, some Poskim even say that one should not wait until Motzei Shabbos to say it, but should rather say it as soon as he has a chance. [See M”B 426:20 and Poskim in previous footnotes]
 Rama 426:2; Terumas Hadeshen 35; Shaar Hakolel 33:1; Mentioned by Rebbe in Sefer Hasichos Noach 1992 regarding being careful with the Zman of Kiddush Levana; However, the Rebbe was seen on numerous occasions to bless the new moon on Motzei Shabbos after the 10th of the month [see Likkut Dinei Rosh Chodesh 12 footnote 32]
 See Rama ibid who writes immediately after this law “and when reciting Kiddush Levana during a weeknight”; When Saturday night falls past the 10th of the month but prior to the 15th then one is often required to say the blessing over the new moon during one of the weeknights being that the previous Saturday night is prior to seven days from the Molad. [see next] If, however, the previous Saturday night is indeed after seven days from the Molad, then certainly one is to say it then, as is accustomed.
Saying it the previous Motzei Shabbos if it is prior to seven days from the Molad: Admur in the Siddur writes that based on Kabala one is to wait seven days past the Molad and hence in general we always wait until after seven days to recited. Practically, however, in the winter the Tzemach Tzedek would say Kiddush Levana even before seven days have passed from the Molad. The reason for this is because of the cloudy and rainy weather experienced in the Russian winter which prevents the moon from being seen. Hence, delaying Kiddush Levana when the opportunity arises prior to seven days could cause one to not say it at all due to the moon’s inability to be seen past the seventh. [Glosses of Rebbe Rashab in Siddur Im Dach p. 321; Shaar Hakolel 33:2] Accordingly, even according to the Chabad custom, in those areas that do not have much visibility of the moon due to cloudy skies , as is common in the winter months, if Motzei Shabbos falls after three days but prior to seven days, Kiddush Levana is to be recited that Motzei Shabbos and is not to be delayed. In cases of question or doubt in this matter one is to follow the ruling of the Rabbinical authority of his community. See Rebbe in Sichas Noach 5752 If one is not stringent one can say Kiddush Levana on Motzei Shabbos that falls after the 3rd day past the Molad, even though it is before the seventh. This especially applies in cloudy areas and particularly in the winter. The Rebbe concludes that: each community should follow in accordance to his level of cloudiness, and it is given to the discretion of the local Rav to decide whether to precede the Kiddush Levana to Motzei Shabbos that is before the seventh of the month. It is implied from the above wording that [even according to the Chabad custom] one may choose to say Kiddush Levana on Motzei Shabbos that is prior to the seventh even if it is not cloudy or rainy. When it is cloudy or rainy one is specifically to do so. However, from footnote 14 there it is implied that in a non-cloudy area one is to wait.
Saying it the previous Motzei Shabbos if it is prior to seven full days from the Molad, but is past 6 days: Some Poskim rule it is not necessary to wait until seven full days pass from the Molad, and rather as soon as six days have passed and one has now entered into the seventh day past the Molad, Kiddush Levana may be recited. Others however rule that based on Kabala one is to wait for a full seven days to pass from the Molad before saying Kiddush Levana. Practically, one may be lenient in this matter. [Rameh Mepuno 78; Shiyurei Kneses Hagedola; Elya Raba 426:14; Shaareiy Teshuvah 426:10; Ashel Avraham Butchach 426:2; So concludes Hiskashrus 454 p. 15 and so was the custom of Rav Yaakov Landau, as told to me by his son Rav Eliyahu Landa Shlita. Seemingly the reason one may be lenient is because even regarding waiting seven days there is a dispute and the Rebbe leaves room to be lenient. See also Sichas 1986 6th Adar Rishon that Kiddush Levana was said that Motzei Shabbos even though seven full days had not passed since the Molad.]
 Rama 426:2; Admur in Siddur; At the very least one should wear the Shabbos Kapata. [Peri Chadash ibid] The Rebbe stated that one is to take special care to recite Kiddush Levana with nice, elegant clothing and that doing so is connected with the final and complete redemption. [Sichas Noach 1992]
 See Q&A!
 The reason: As perhaps there will be cloudy skies on that Motzei Shabbos and a few days onwards, such as two, three or four cloudy nights [depending on when Motzei Shabbos falls, if on the 11th, 12th, or 13th]. Due to this the moon will not be visible and the time for saying Kiddush Levana will pass. [Rama ibid; Terumos Hadeshen ibid] Thus, even according to those who hold that fulfilling a mitzvah later but with greater Hiddur is more important than fulfilling it earlier but with less Hiddur, would agree that in the case that there is question of whether the mitzvah can be fulfilled at a later date then certainly one is to fulfill at an earlier date even if it will be a lesser level of Hiddur. [Terumos Hadeshen ibid; See Admur 25:4 that Zerizin Makdimim overrules delayed Hiddur, 94:5 that later hiddur overrides Zerizin Makdimim; Likkutei Sichos 19 p. 76 in footnote 73 that both rulings are correct, as in 25:4 it discusses an external Hiddur of preceding the Tallis which is Tadir, and in 94:5 it discusses a Hiddur in the actual Mitzvah]
The law if one is aware of the weather forecast: Vetzaruch Iyun if the above law which requires preceding the recital of the blessing to prior to Saturday night in the event that it falls past the 10th of the month applies even if one is aware of the weather forecast and cloudy skies are not expected on Saturday night or onwards, as in such a case the whole premises of the ruling falls as the whole basis of the ruling of the Terumas Hadeshen is due to the question of whether it will be cloudy. Vetzaruch Iyun as to how much one can actually rely on weather forecast regarding cloudy skies. Nonetheless, perhaps the above can be a Limud Zechus for those who don’t abide by the above law and arrange Kiddush Levana for Motzei Shabbos even when it is past the 10th of the month. However, if in truth they do not know the weather forecast and there is seemingly no reliance for them to do so.
The law for those who permit saying Kiddush Levana with a blessing past the 15th of the month: Tzaruch Iyun if such people may delay saying it until Motzei Shabbos if they will remain with 4 days according to their accustomed end date.
 M”A 426:3; Brought in M”B 426:6; Kaf Hachaim 426:23; Although the Rama writes “When Motzei Shabbos falls before the 10th of the month” nevertheless based on his reason of suspecting for up to four cloudy nights and not more, it is evident that he means to also include the 10th. [M”A ibid]
 This can occur if the Molad fell on the 29th or on the night of the 30th day, and Rosh Chodesh was on the 31st.
 M”A 426:3; As according to the Rama one may only say Kiddush Levana up until 14 days 18 hours and 396.5 chalakim past the Molad, hence one must calculate that Motzei Shabbos is not the 4th to last night to say Kiddush Levana, as in such a case we suspect for four cloudy nights.
 Ashel Avraham Butchacher 426:2; Biur Halacha 426:2 “Ela”; Kitzur Shulchan Aruch 97:9; Daas Torah 426; See Kaf Hachaim 426:23 “If the time for blessing the moon has arrived and one is unable to do so with the public, who he is nonetheless not to delay due to this” See Likkut Dinei Rosh Chodesh 12 footnote 64
 The reason: Saying the blessing of the moon together with a minyan is not a halachic requirement and is not even recorded in the Shulchan Aruch. It is certainly of no greater value than is the recital of it on Motzei Shabbos which is recorded in the Shulchan Aruch. Accordingly, if we rule that one should not delay until Saturday night to bless the moon if it will be past the 10th day of the month, then certainly when should not delay past the 10th day of the month in order to say it with a minyan.