Talking during Kerias Hatorah:
Once the reader has begun to read the Torah it is forbidden for the congregants to speak. [Furthermore, some Poskim rule that even prior to the start of the reading, once the Sefer Torah is opened, it is forbidden to speak.]
Between Aliyos: It is forbidden to speak even between the Aliyos, [even when the MiShebeirah is recited, and hence from the commencing of the reading until the conclusion of the final reading, speech is prohibited]. This applies even to words of Torah. It is however permitted for one to be lenient to learn Torah quietly to himself in between the Aliyos. However, once the blessings are recited he must be quiet and listen.
Michaber 146/2; Sotah 39a; Zohar Vayakhel p. 206
 This prohibition applies according to all opinions regarding mundane words, as it is only regarding words of Torah in which there is a dispute.
 M”A 146/3 based on Gemara ibid; Gr”a; Soles Belula 146/3; M”B 146/4
Other opinions: Some Poskim rule the prohibition of speech only begins from the time that the reading commences. [Michaber ibid as explained in Beis Yosef; Rashi on Gemara ibid; Elya Raba 146/3; Mateh Yehuda146/3; Derech Hachaim 3; Shaareiy Efraim 4/11; Kaf Hachaim 146/8]
 1st opinion of Michaber ibid and his final ruling; Peri Chadash; Shelah 77, brought in M”A 146/5; Gr”a in Maaseh Rav; Aruch Hashulchan 146/6; M”B 146/15; Achronim as brought in Kaf Hachaim 146/13; Rebbe Rashab in Modaah Raba [printed in Siddur ImDach p. 644]
Background and other opinions in Michaber: The Michaber records five opinions regarding learning Torah during the reading. 1) It is forbidden to do so in any circumstance; 2) Some opinions [Rashi] permit one to learn quietly. 3) Some opinions [Bahag] rule that if there are ten other people listening to the reading, it is permitted for one to speak [words of Torah-Rama]. 4) Some opinions [Rif] rule if one’s occupation is Torah learning he may learn Torah. 5) Some opinions [Talmidei Rabbeinu Yonah] rule that it is permitted for one to learn Torah during the reading if prior to the opening of the Torah he turned his face away from the Torah and made himself appear that he does not desire to hear the reading but rather to learn, and he began to learn. In the Beis Yosef 146 he concludes that it is valid to rely on all of the above Sages. However the Michaber ibid concludes “The proper ruling to follow is that by all the Parshiyos it is befitting for a meticulous Jew to pay attention and listen to the reading.” The Rebbe Rashab in Modaah Raba [printed in Siddur Im Dach p. 644] explains that based on the Kelalei Haposkim it is evident that the Michaber rules like the first opinion. So is also the ruling of all the Achronim to be mentioned below that one may only be lenient to learn quietly between the Aliyos. [Kaf Hachaim 146/13] The Shelah ibid concludes that the Torah Scholars that learn Torah during the reading indirectly cause others to speak during the reading and will be punished for doing so.
Parshas Zachar and Parah: According to all opinions the reading of Zachar and Parah must be listened and heard by the congregants [even by one who desires to learn Torah based on one of the above allowances], as the reading of these Parshiyos with a Minyan are Biblical. [Michaber ibid]
Olas Tamid 146/3; Beis Yosef in name of Rashba; Kaf Hachaim 146/12
Other opinions: Some Poskim rule it is permitted to speak even mundane words between the Aliyos. [Peri Chadash 146/2; Bach as brought in Beir Heiytiv 146/3; P”M 146 A”A 7]
Mareh Yichezkal 26; Kaf Hachaim 146/15
A long MiShebeirach: In today’s times that the MiShebeirach can take quite some time, some justify those that are lenient to speak necessary words during this time, even if they are of mundane content. This however only applies to a short interjection that will not turn into a long conversation that will interfere with the reading. [See other opinions in previous footnote; Aruch Hashulchan 146/3; Shulchan Hatahor 146/1; Piskeiy Teshuvos 146/3] The Kaf Hachaim 146/10 completely negates their opinion.
Beis Yosef 146
The reason: As we suspect the person will continue his conversation even after the reading resumes. [Beis Yosef; M”B 146/6]
Drashos Maharil; Kneses Hagedola 146/1
Other opinions: Some Poskim rule it is permitted to speak words of Torah between the Aliyos, and that so is the custom. The reason for this is because the entire reason for the prohibition during the reading is in order not to disturb others, and hence in-between the reading there is nothing to suspect for. [Bach 146, brought in M”A146/3; M”B 146/6; Peri Chadash 146/2] According to this opinion it is permitted for one to learn even aloud. [Kaf Hachaim 146/10]
 Seder Hayom; Elya Raba 146/4; M”B 146/6; Chesed Lealafim 146/13; Kaf Hachaim 146/10 and 13
Other opinions: From the Maharil ibid it is implied that he forbids learning even quietly to oneself.
Elya Raba ibid