A. By which blessing?
The Sages instituted to blow a set of blows [after concluding] the blessing of Malchiyos, a set of blows [after concluding] the blessing of Zichronos and a set of blows [after concluding] the blessing of Shofros. [Thus in total the Sages instituted that the blows take place at three different times within the Musaf prayer.]
B. By which prayer of Musaf? Is the Shofar blown both during the silent prayer and the repetition of Musaf?
The institutional blowing of the Shofar during Musaf is to take place during the Chazan’s repetition and not during the silent prayer of the congregation. However some communities have the custom to blow twice, once within the silent prayer and once during the Chazan’s repetition. Nevertheless a community that does not have such a custom [to blow also during the silent prayer] should not initially take upon themselves this custom. [Practically the Chabad custom is to blow both within the silent prayer and within the Chazan’s repetition. This is also the custom of many Chassidic communities as well as many Sefaradi communities. However in majority of the Ashkenazi communities the custom is not to blow during the silent Shemoneh Esrei.]
The custom of Chassidim is to blow Shofar both within the silent prayer and the Chazan’s repetition.
Q&A on the Tekios in the silent Shemoneh Esrei
How is the Chazan to motion to the Baal Tokeia that he has finished the blessing?
He is to bang his hand on the Amud. He is not to finish the blessing out loud.
What is one to do if he has not reached the end of the blessing in his silent Shemoneh Esrei and the blowing has begun?
One is to stop praying and listen to the Tekios. [It is not necessary to Daven quickly in order to reach the end of the blessing together with the Chazan.]
What is one to do if he has already reached the Tekios in his silent prayer and the blowing has not yet begun?
One is to stop and wait until the Tekios are blown.
If the Baal Tokeia made a mistake in the blowing during the silent Shemoneh Esrei is one to correct him?
Sparks of Kabala:
The blowing of Shofar during the silent Shemoneh Esrei is a great Segula to help subjugate the evil inclination involved in sins of promiscuity and Giluiy Arayos. It is for this reason that the Chassidim held on to this custom with much fervor.
 592/1; Michaber 592/1
 592/1; See Avnei Nezer 445-447; Minchas Elazar 4/37; Mishmeres Shalom 41/7
 Admur ibid; Michaber 592/1; Shulchan Gavoa 591/2; implication of Tur; Implication of majority of Poskim that omit any mention of blowing during the silent prayer; See Kaf Hachaim 592/1
The reason: The Sages instituted to blow the Shofar after each of the three blessings recited by the congregational prayer of Musaf. Now since in the silent prayer of Musaf everyone prays to themselves and hence everyone is holding in a different area of the prayer and by a different blessing [we are therefore unable to fulfill the institution of blowing after the blessing of Malchiyos, Zichronos and Shofros]. Thus we do not blow the Shofar at all within this prayer just like we do not blow the Shofar when praying without a Minyan and rather we delay the blowing until the Chazans repetition. [ibid; Meaning that just like when a Minyan is not present the Shofar is not blown due to that the institution was to only blow with a Minyan, therefore also by the silent prayer we do not blow as it is considered to be as if a Minyan is not present being there isn’t a Minyan holding by the same blessing.]
 This custom is recorded in Admur ibid; Kneses Hagedola 592; M”A 592/1; Tur 592; So rule the following Poskim: Hearuch “Meiah Piyos” [that so is their custom]; Hamanhig R”H 21; Tosafus R”H 33b in name of Aruch; Arizal in Shaar Hakavanos p. 99; Peri Eitz Chaim 26/3; Shlah p. 217 [“the Arizal revealed the importance of this custom and that it accomplished great Tikkunim]; Kaf Hachaim 585/28; 592/1; See Yeshuos Moshe 1/41 that defends this custom saying that since today we recite Piyutim in the repetition therefore the prayer is considered invalidated and hence the main time to blow the Shofar is by the silent Shemoneh Esrei.
The reason: The reason behind the custom to blow also during the silent prayer is in order to accumulate 100 blows in correspondence to the 100 sounds of cry that the mother of Sisrei sounded. Thus 30 are sounded before Musaf, 30 during the silent prayer of Musaf, 30 during the repetition of Musaf and ten after Musaf. [Hearuch; Hamanhig; Tosafus R”H 33b; brought in Kaf Hachaim 592/1]
 Admur ibid; Kneses Hagedola 592; M”A 592/1
 Admur in Siddur; Avnei Nezer 447 that Admur wrote the SH”A according to the Poskim however he himself followed the custom of the Baal Shem Tov and the Maggid to blow by the silent prayer and hence he ruled that way in the Siddur.
 Alef Hamagen 591/11; Avnei Nezer 445-447 [although he himself discourages the custom]; Minchas Elazar 4/37; Mishmeres Shalom 41/7
Background of the Chassidic custom:
This custom is based on the Arizal, and was followed by the Baal Shem Tov and his students including the Admorim of Kotzk and Gur. [see Avnie Nezer ibid] Hence in practically all Chassidic communities the custom is to blow during the silent Musaf prayer. However the Avnei Nezer [445/446] discouraged this custom saying it is not something which is relevant for all the Chassidim to follow, and he himself was not accustomed to do so by his Minyan on R”H. He writes mockingly “The young Chassidim think that blowing Shofar in the silent Shemoneh Esrei is a foundation of Chassidic thought”. The Minchas Elazar [4/37] however sharply criticizes this ruling of the Avnei Nezer and upholds the long standing custom of Chassidic communities to blow Shofar during the silent Musaf prayer saying that it certainly is a foundation of Chassidic thought as it has a Segula to nullify the Yeitzer Hara of Zera Levatala.
 As is the ruling of Kabalas as stated above from the Arizal and so is the custom of the Beis Keil community in Jerusalem as well as other communities. [Kaf Hachaim 592/1] So is the Sefardi custom. [Piskeiy Teshuvos 592/2; Yechaveh Daas 2/75] Regarding the opinion of the Yaavetz: See Avnei Nezer 447 that says he writes not to blow while others say he wrote that one is to blow. However it is difficult to recognize what parts of the Siddur were written by the Yaavetz and which were added by the editors. Vetzaruch Iyun.
 M”E 591/13 [That only unique individuals that follow the teachings of Kabala are accustomed to do so. However see the Alef Hamagen 591/11 which writes that in these communities [Poland and Gelitzia] most are accustomed to do so.]
 Minchas Elazar 4/37
 Mateh Efraim 591/13; Kaf Hachaim 592/2
 See Avnei Nezer 445 that both the Admur of Kotzk and Gur Davened at a slower pace than the Chazan and they did not motion him to wait for them.
 Mateh Efraim 591/13; See Tiferes Adam 3/8 regarding if one will be waiting so long that he could have Davened the entire Shemoneh Esrei in the interim.
 Mateh Efraim 591/13; Kaf Hachaim 591/6; 592/2; Rashba 295
 Kaf Hachaim 591/6
 Arizal brought in Minchas Elazar 4/37
 Minchas Elazar ibid