Reading Hashem’s name in a verse in the Mishneh, Talmud, or other learning text:
Some Poskim rule that when reading the Talmud [or other Torah Sefer] one may read Hashem’s name in the verses that are found in the text. [This applies even if the verses are not complete.] Furthermore, some Poskim rule one is actually obligated to recite Hashem’s name. Other Poskim, however, rule that one is to refrain from mentioning Hashem’s name and is rather to say Hashem or Elokeinu or Shakaiy and the like, and so was seen to be the Rebbe’s custom. One however is not to say the term, Adoshem, despite the common practice.
Reading Hashem’s name in a blessing in Talmud or Poskim:
When reading the Talmud [or other Torah Sefer] one may not read Hashem’s name in the blessings that are found in the text. [Some Poskim rule that the same law applies towards a child. It is however permitted for a child to do so for education purposes, in order to properly learn the blessing.]
Saying Hashem’s name during a speech:
When quoting a verse in a speech one may recite Hashem’s name, just as is the law regarding one who is reading from a text. However, as stated above, some refrain from doing so.
 Sheilas Yaavetz 81, brought in Shaareiy Teshuvah 215:4; Chesed Lealafim 215:7; Pela Yoeitz Shem Shamayim; P”M 215 A”A 5; Torah Leshma 496; Shoel Umeishiv Kama 3:53; Hisorerus Teshuvah 1:4; Toras Chaim Sofer 215:5; M”B 215:14; Maharil Diskin KU”A 104; See Or Letziyon 2:14-36; Yabia Omer 3:14; Yechaveh Daas 3:13; Tzitzi Eliezer 13:1; Mishnhe Halachos 4:27; Piskeiy Teshuvos 215:10
 Igros Moshe 2:56 as is implied from Sheilas Yavetz ibid; See Chayeh Adam 5:2; Piskeiy Teshuvos 215:10
 Sheilas Yaavetz ibid in name of Chacham Tzevi and so is his opinion and so was custom of Rabbanim that he witnessed; Daas Torah 215 in name of Zechor Leavraham in name of Shelah; Igros Moshe 2:56 regarding a whole verse that one must say Hasehm’s name; See however P”M and M”B ibid who write “He is permitted to say it”, See however Igros Moshe ibid who interprets them to refer to a partial verse.
 The reason: As it is not befitting to recite Hashem’s name in its unnatural form. [See Taz 621:2] Alternatively, this is because it is forbidden to stop in the midst of a verse in an area that Moshe did not stop, and if one does not say Hashem’s name it is considered as if he stopped in middle of a verse. [Igros Moshe 2:56 in his reason behind the ruling of the Yaavetz]
 Tashbatz 419 in name of Maharam that so was his custom; Ikarei Daat 1:21 based on Maharsha Sanhedrin 114b; Zichronos Eliyahu of Rav Eliyahu Mani Mareches Mem based on Tashbatz ibid; Aruch Hashulchan 215:2 that so is custom even though there are Poskim that are lenient
 Taz 621:2; Yosef Ometz 349; See Piskeiy Teshuvos 215 footnote 58
 M”A 215:5; Birkeiy Yosef 215 that so is custom of elderly Rabbis; Machazik Bracha in name of Tashbatz Katan 419 in name of Maharam, brought in Shaareiy Teshuvah 215:4; P”M 215 A”A 5; M”B 215:14; Igros Moshe 2:56
Other opinions: Some Poskim rule it is permitted and an obligation to recite Hashem’s name even in the blessings that are found in the text and that one is specifically to do so in order so he learn the correct dialect [Sheilas Yaavetz 81] See also regarding Kiddush Levana that the following Poskim rule one may read the blessing in the Talmud:Tur with Hashem’s name even past the time: Divrei Torah 4:107; Darkei Chaim Veshalom 495; See also Avnei Nezer 383; Taamei Haminhagim p. 199 brings from Derech Pekudecha in name of the Chozeh Melublin that after the time has passed one may nevertheless read the blessing from the Gemara. Alef Lamagen 581:22 concludes from Derech Pekudecha that so is the custom of the world to say it from a Gemara with a blessing after the correct time; Mentioned in words of Rebbe Rashab in Shemuos Vesipurim p. 184
 Igros Moshe 2:56
 Admur 215:2