Not to talk of irrelevant matters until after Musaf:
One must beware not to talk of irrelevant matters between the Tekios Meyushav [which is the first set of blows that is sounded after Kerias Hatorah] and the Tekios Meumad [which is the second set of blows in Musaf]. One is not to talk until after the completion of these blows sounded in Musaf. This applies for both the Baal Tokeia, who said the blessing over the Shofar, and the congregation, who did not say a blessing over the Shofar but heard it from the Baal Tokeia, nevertheless, even they may not talk until after the Tekios Meumad.
Matters relating to the Tekios or prayer: It is only initially forbidden to speak of irrelevant matters between the two sets of blows. However, it is even initially permitted to speak of matters relating to the prayer or blowing.
Bedieved if one talked: If one transgressed and spoke between the Tekios of Meyushav and Meumad, he nevertheless is not required to repeat the blessing over the blows of Meumad. This applies for both the Baal Tokeia and the listeners. This applies even if he spoke of irrelevant matters.
Talking between the blessing and the blowing: See Chapter 2 Halacha 9C
Talking between blows of Meyushav: See Chapter 3 Halacha 9!
One may not talk of irrelevant matters from the time he hears the beginning of Shofar blowing after Kerias Hatorah until after Musaf. However one may speak of matters relating to the Shofar and Davening .
Q&A on not speaking until after Davening
May one say Asher Yatzar after using the bathroom?
If one sees lightning or hears thunder prior to the conclusion of Musaf may he say the blessing?
May one say Tehillim or learn Torah during this time?
It is forbidden for one to learn Torah verbally until after Musaf. This applies even prior to Chazaras Hashatz. However it is permitted to think Torah in one’s mind. One may however even verbally learn matters of Torah that relate to the Tekios. Some also permit one to say Tehillim during this time.
May one talk after the blowing of Musaf, prior to hearing the extra blows after Musaf?
It is proper to refrain from talking unrelated speech until after the finals blows are sounded after Davening. Hence one is to refrain from talking until after the 30 blows are sounded after Tehillim. However from the letter of the law one is permitted to talk of even unrelated matters once the blows of Musaf have been sounded.
 592/7; Michaber 592/3
 The reason: The reason for this is because the blessing said over the 1st set of blows includes also the second set of blows, [and it is forbidden to make an interval between a blessing and its Mitzvah]. [Admur ibid; Taz 592/2; Biur Hagr”a; M”B 592/8] Now although in truth the blows of Meyushav and Meumad are considered like a single Mitzvah [see below] nevertheless since it appears like two Mitzvos he may not initially speak of irrelevant matters. [Admur ibid]
The Piyutim: The Piyutim that are read during the repetition are not considered an interval as they relate to the prayer. [Chayeh Adam 141/9; Kaf Hachaim 692/177]
The Yehi Ratzon: Some Poskim rule that the Yehi Ratzon [and Viduiy] that some recite between the sets of the Tekios Meyushav is considered an interval. [Derech Hachaim; Tosefes Chaim on Chayeh Adam ibid; M”B 592/12] Others however argue that the Yehi Ratzon is not an interval. [Olas Shmuel 88; Siddur Yaavetz; Shaareiy Tziyon 592/15; Kaf Hachaim 592/17] The Chabad custom is not to recite the Yehi Ratzon between the Tekios [Igros Kodesh 3/196] however the Yehi Ratzon after the Tekios is recited. [See Otzer Minhagei Chabad 298]
 Since they fulfilled their obligation of the blessing through hearing the blessing of the Baal Tokeia it is therefore considered as if they said the blessing and hence they too are not to speak in-between. [Birkeiy Yosef 592/2; Kaf Hachaim 592/15]
 Admur ibid; Rama 592/3; Darkei Moshe 592/2 in name of Mahril; M”A 592/3; M”B 592/14
 The reason: As in truth the two sets is all considered a single Mitzvah [as explained in the next footnote] [ibid] Perhaps one can suggest that the explanation is as follows: Since they appear like two different Mitzvos therefore one should not speak in between as this can teach people to speak between two Mitzvos and nevertheless not repeat the blessing. However this suspicion only applies by irrelevant matters, as only irrelevant matters cause the blessing to be repeated between two Mitzvos. However relevant matters are valid Bedieved even between two Mitzvos [See Admur ibid as well as Admur 167/9; 206/3; Seder 9/1; Ketzos Hashulchan 37/7] and hence there is no need here to refrain from talking of relevant matters, as a) in truth it is one Mitzvah and hence even initially allowed and b) even if the onlooker will learn to do this between two Mitzvos, in any event it is valid Bedieved. Vetzaruch Iyun!
 Admur ibid; Rama 592/3
 The reason: Although one who recites a single blessing on behalf of two Mitzvos is required to repeat the blessing prior to the second Mitzvah if he spoke in between, as explained in 8/21 [regarding speaking between two pairs or Tzitzis] and 25/18 [regarding speaking between the Yad and Rosh Tefillin] nevertheless since the blows of Meyushav and Meumad are considered like a single Mitzvah therefore if one transgressed and spoke between them even of irrelevant matters he nevertheless is not required to repeat the blessing, as an interval never invalidates the blessing when one is still in the midst of the same Mitzvah. This is clearly seen in the law of one who speaks while in a Sukkah or while wearing Tefillin in which case it is not considered an interval even though every moment that he is sitting in the Sukkah and wearing Tefillin he is fulfilling the Mitzvah of Hashem. Furthermore even initially he may speak while in the Sukkah and wearing Tefillin, however here by Shofar since it appears like two Mitzvos he may not initially speak of irrelevant matters. [Admur ibid]
The reason the two sets of blows are considered one Mitzvah: The institution of the Sages to blow in Musaf did not obligate the congregation to blow the Shofar on two different occasions, before and after Musaf, but rather merely that they should fulfill their Biblical obligation of Shofar blowing within the Musaf prayer. Nevertheless the custom of all Jewry became to blow two sets of blows, before Musaf and during Musaf; before Musaf in order to fulfill the Biblical obligation and during Musaf in order to fulfill the institution of the Sages. The reason for this is because we desire to confuse the heavenly prosecutor prior to the Musaf prayer in order so he does not prosecute during the Musaf prayer, as fulfilling the Mitzvah of blowing the Shofar enters the Satan into a state of confusion and distress. [Thus in order so the blowing within Musaf does not interfere with the desire to confuse the Satan prior to Musaf the custom evolved to establish two occasions of blowing Shofar.] Now, we say the blessing prior to the first set of blows which fulfills the Biblical obligation and then continue to perform the Rabbinical obligation to blow the Shofar within the order of the blessings recited in Musaf as the Sages instituted. [ibid] [Seemingly the explanation of this is as follows: The Sages did not simply institute to blow Shofar during Musaf, they instituted that one is to fulfill his Biblical obligation of the Tekios in Musaf. Now, this created a Rabbinical form of Biblical Tekios, Tekios that must be heard during Musaf, as the actual Biblical Tekios can be heard at any time. Thus when one decides to blow the Biblical Tekios before Musaf he has not yet fulfilled his Rabbinical obligation of the Biblical Tekios. For this reason it is all considered one Mitzvah as from a Rabbinical perspective the Biblical Mitzvah that was started before Musaf is not completed until it is blown within Musaf and it is hence all a continuation of the same Biblical Mitzvah. This answers a number of queries on the above explanation: a) How can we uproot the decree of the Sages and precede the Biblical Tekios to before Musaf? b) How is it considered a single Mitzvah of one is Biblical and the other is Rabbinical? C) Why we would think that Musaf invalidates the Biblical Tekios and thus need a special Halacha in 593/4 to teach us otherwise.]
 Minchas Yitzchak 3//44; 4/47; Sheivet Haleivi 5/66; Tzitz Eliezer 11/45; Kinyan Torah 5/56; Piskeiy Teshuvos 592/4
The reason: As it has the same status as Pesukei Dezimra.
 Daas Torah 592
The reason: As the time between the two Mitzvos has the same status as Birchas Shema.
 Piskeiy Teshuvos ibid; See Admur 66/4; This would apply even if the two sets of blows have a status of Birchas Shema.
 Har Tzevi 1/42; Piskeiy Teshuvos 592/4
 Leket Yosher from the student of the Terumas Hadeshen p. 128; brought in Minchas Yitzchak 4/48
 Yesod Veshoresh Havoda 11; Piskeiy Teshuvos ibid; however perhaps there he is only referring to Chazaras Hashatz of Shacharis, Vetzaruch Iyun; See however Otzer Minhagei Chabad p. 128 that the Rebbe at times would say Tehillim during the Chazan’s recital of Hineni Heani. Seemingly the reason this is not considered an interval is because it is similar to matters relating to prayer which one may even initially speak of during this time.
 Chayeh Adam 141/9; Kaf Hachaim 592/17; The Rebbe’s custom was to refrain from talking until after the last Tekios sounded after Davening. Only then would he wish the Kohanim a Yasher Koach. However in the year 1992 the Rebbe changed from his custom and recited Yasher Koach even before the last blows. [Otzer Minhagei Chabad 340]
The reason: It is proper not to do so in order so the blessing cover also these blows.
 Kaf Hachaim ibid
 592/7 that the prohibition is only between the Tekios Meyushav and Meumad; 590/19 that these last Tekios don’t even have to be redone if they were invalid.