Gemara

We cannot burn a Mezuzah - "Lo Sa'ason Ken la'Shem Elokeichem."

Shabbos 61b (Beraisa): if Hash-m's name was written on the handle of a Keli or the leg of a bed, one cuts off and buries the part containing His name.

104b (Beraisa - R. Yehudah): If a scribe needed to write Hash-m's name in a Sefer Torah, but he intended to write 'Yehudah' and forgot the Dalet, he may pass the quill over the letters, to be Mekadesh the name;

Chachamim say, this is not an ideal writing of Hash-m's name.

Shevuos 35a (Beraisa): One may not erase Kel, Eloka, Elokim, Elokeichem, Echyeh Asher Echyeh, 'Aleph-Dalet', 'Yud-Kei', Shakai, and Tzevakos. One may erase ha'Gadol, ha'Gibor, ha'Nora... Chanun, Rachum....

35b (Beraisa): Once one writes 'Aleph-Lamed' from Elokim, he may not erase it. The same applies to 'Yud-Kei' of Hash-m's name. One may erase 'Shin-Dalet', 'Aleph-Dalet', or 'Tzadi-Beis' from Shakai, Adon-i, or Tzevakos;

R. Yosi says, the entire word 'Tzevakos' may be erased, for it is not a name of Hash-m. It refers to Yisrael - "Tziv'osai Es Ami Vnei Yisrael."

(Shmuel): The Halachah does not follow R. Yosi.

(Beraisa): If 'Lamed', 'Beis', 'Vav', 'Mem', 'Shin', 'Hei', or 'Kaf' is prefixed to Hash-m's name, it may be erased. If 'Nun-Vav', 'Hei-Mem', or 'Chaf-Mem' was suffixed to 'Elokei', it may be erased;

Others say, the suffixes may not be erased, because Hash-m's name that precedes them makes them Kodesh.

(Rav Huna): The Halachah follows this latter opinion.

Rishonim

Rambam (Hilchos Yesodei ha'Torah 6:2): One who erases a Kodesh name of Hash-m is lashed mid'Oraisa. It says about idolatry "v'Ibadtem Es Shemam... Lo Sa'asun Ken la'Shem."

Hagahos Maimoniyos (1, citing the Re'em): If one wrote Hash-m's name without intent to be Mekadesh it, it is not Kodesh. If a scribe intended to write 'Yehudah' and forgot the Dalet, the name is not Kadosh.

Rosh (Teshuvah 3:15): One may not say 'Shalom' in a bathhouse (Shabbos 10a), but we do not find anyone who forbids erasing it. People write it in letters, and it is thrown out.

Tosfos (Sotah 10a DH Ela): One may not say Shalom in a bathhouse. Also, one may not erase it. The Tana (in Maseches Sofrim Perek 4) lists names that one may not erase; he omitted Shalom.

Poskim

Shulchan Aruch (YD 276:9): One may not erase even one letter of the seven names of Hash-m that may not be erased. One may not erase their suffixes, e.g. the final 'Chof' in Elokecha or 'Chof Mem' in Elokeichem. The names are Shem Havayah (Yud 'Kei' Vov 'Kei), Shem Aleph-Dalet (that ends 'noy'), Kel, Eloka, Elokim, Shakai, and Tzevakos. Some texts add 'Echyeh Asher Echyeh.'

Bedek ha'Bayis: Orchos Chayim says that Rav Sadya Gaon says that if a scribe wrote a Kinuy in the wrong place, he may erase it. The Ra'avad says that if he did not err, or even if he erred but he wrote the entire verse, he may not erase it. It seems that the Rambam and Tur disagree.

Shulchan Aruch (11): If a letter (of a Sefer Torah) stuck to another in Hash-m's name, he can scrape it (the ink joining them) away.

Rema: If ink spilled on Hash-m's name, one may erase it, in order to fix it. This is not like erasing (Hash-m's name). It is like fixing. If in a Hei in Hash-m's name the leg properly touches the roof and it looks like a Ches, one may erase and scrape the lefg to fix it. If it touches the roof only a little, it is not clear whether it is permitted.

Terumas ha'Deshen (72, cited in Beis Yosef DH Yesh Bo): May one fix a 'Hei' of Hash-m's name (e.g. if the legs touch)? Hagahos Maimoniyos (Hilchos Yesodei ha'Torah 6:2) permits to erase a Hei into which a drop of ink fell, for he intends only to fix it. It seems that the same applies here. Or, perhaps there it is permitted because the entire letter was illegible, but here we merely need a minor correction. Or, perhaps here it is permitted, for one must remove only the leg written improperly, but he will not touch what was written properly. The Tur brings Ri Alexandri, who permits. Lo Sa'asun Ken does not apply. If ink spilled on Hash-m's name, one may erase it, for he intends only to fix. If one may erase a letter that was written properly and became ruined, all the more so one may fix a letter that was never written properly! If the leg of the latter Hei touches the roof and it became a Ches, even though the first half of Hash-m's name was Mekadesh it, one may fix it. If a tiny strand connects the leg to the roof, and it is read like a Hei, the Rosh (Hilchos Sefer Torah 12) says that in Tefilin there is no solution, for they must be written in order. Shabbos 104b says that if one removed part of a letter and changed it to another letter, this is called correcting. It is as if he erased the initial letter. The same applies to erasing the leg of the Hei.

Shulchan Aruch (12): If one needed to write Hash-m's name, and wrote 'Yehudah', he makes the Dalet into a Hei, and erases the latter Hei.

Taz (7): Why may he erase the Hei? We should say that the initial 'Yud-Kei' is Kodesh, and it is Mekadesh whatever comes after! We say so only when it has a meaning, e.g. Elokeichem. - your G-d. This is why the Beraisa mentioned particular suffixes that Hash-m's name is Mekadesh, and did not merely say that it is Mekadesh letters appended at the end.

Rema (13): L'Chatchilah one may not write Hash-m's name not in a Sefer, lest it come to disgrace. Therefore, we do not write it in letters. Some are careful even not to finish the word Shalom.

Shach (16): Most people are not careful about Shalom. The Rosh (Teshuvah 3:15) permits.

Nekudas ha'Kesef: However, Tosfos forbids. Some say that this is only to explain the question in Sotah. I say that he truly forbids. Other Poskim do not say that it is forbidden. In practice, this requires investigation.

Pischei Teshuvah (28): The Radvaz (Chadashos 202) says that one must be careful about a greeting with Shalom, for he gives Shalom from Ba'al ha'Shalom (Hash-m). When it means 'peace', it has no Kedushah.

Chavos Ya'ir (16): A Chacham permitted to stamp Hash-m's four-letter name with a signet ring, for this is not writing. He assumes that one may write Hash-m's name only in its place, i.e. in a Sefer Torah, Tefilin, Mezuzah, Berachos, etc. The Rema connotes like this. However, there we discuss only in paper and in letters of Reshus (not Mitzvos), where it is prone to come to disgrace. This does not apply to a ring. It is improper to write Hash-m's name without need, lest it come to disgrace, and if one did, it must be buried, but it is not an actual Isur to write it. One who stamps Hash-m's name transgresses Lifnei Iver if the recipient will throw away the paper. Also, sometimes it does not come out so clear the first time, and he stamps again over Hash-m's name, and transgresses Lo Sa'asun Ken.

Chavos Ya'ir (16): Even though the one who made the stamp did not intend for Kedushas ha'Shem, it is Kodesh by itself. Even what a Nochri carved out gets Kedushah! Azkaros (names of Hash-m) in a Sefer Torah require Kedushas Peh (the scribe says that they should be Kodesh), but to be stringent to forbid benefit from the place and erasing them, it is Kodesh automatically due to the letters. This is like a firstborn male Tahor Behemah. It is a Mitzvah to be Mekadesh it, but if one did not, it is Kodesh by itself. Why didn't Hagahos Maimoniyos say that the Rambam argues with Re'em? Perhaps the Re'em discusses only what is needed for a Kosher Sefer Torah. We can give a better answer. Why did the Re'em bring a proof from one who intended to write Yehudah, which Tana'im argue about? He should have brought a proof from a Sefer Torah in which the Azkaros were Lo Lishmah. We say that it is not worth anything (Gitin 54b)! Rather, it is worthless, but the Azkaros are Kodesh, for it was written Stam, and Stam they are destined to be written Lishmah. If so, what was written without Kavanah is not Kodesh. This is unreasonable. Rather, we are stringent to say that it has Kedushah. A Nochri write with intent for Emunas Yisrael. This is why the Bach says that b'Di'eved, if a Yisrael told a Nochri to tan parchment for a Sefer Torah, it is Kosher even if the Yisrael did not help. We do not say so when there was explicit intent for Chol, e.g. to write Yehudah.

Chazon Ish (Yadayim 8:18): The Isur to erase Hash-m's name applies to anything written with intent for His name. It does not need intent to be Mekadesh it. We learn from handles of Kelim (Shabbos 61b). Perhaps intent to be Mekadesh applies only to Tanach, in which the custom is to be Mekadesh Azkaros like in a Sefer Torah. However, there must be intent for Hash-m's name. If he intended to write Yehudah, it is not Kodesh. He was Mis'asek (engaged in something else). Just like certain names can be Kodesh or Chol, e.g. Adnus and Elokim, also regarding all names, if one merely wrote letters without intent for a name, it is not Kodesh, just like one who intended to write Yehudah. Maseches Sofriim permits erasing it. The same applies to Hash-m's name written by an Apikores.