The Halachic debate and practical directive:
Does the abbreviation of B”H contain holiness? Some Poskim rule that the abbreviation of B”H does not contain holiness, and thus may be discarded regulalry. Other Poskim rule that the words B”H contain some level of sanctity, being the Hei is a letter of Hashem’s name and is reffering to it.
The various customs for and against: Due to the above latter opinion, some Gedolei Yisrael voiced oppossition to the custom of writing B”H on letterheads due to worry of its sanctity getting desecrated. However, writing BS”D or בעזהש”י as opposed to B”H avoids the issue of desecration, and is hence permitted even according to the previous mentioned Poskim who negate the writing of B”H. Accordingly, many today write BS”D, or another alternative, rather than B”H to avoid this issue, even though the original custom was to write B”H. Others do not write anything at all, and so was practiced by some Geodlei Yisrael. The accepted practice, however, is to continue with the old custom and write B”H and so was the custom of many Gedolei Yisrael, and so was the custom of the Baal Shem Tov and the vast majority of his students, including the Chabad Rabbeim. However, interestingly, on the heading of Mamarim the Rabbeim would write BEZ”H or BS”D and not B”H, which was reserved only for the heading of letters that they wrote.
Practical directive of the Rebbe: Practically, the Rebbe instructed people to write B”H on all forms of letters, although also mentions the alternative of BS”D or BEZ”H. [The Rebbe’s custom was to write B”H on the letterhead of all letters, while BS”D was written on the letterhead of Mamarim.] The Rebbe instructed that this applies even towards a Torah publication [such as a Torah journal, Kuntrus and even Likkutei Sichos pamphlets], that it should nevertheless begin with the above abbreviation of B”H.
It is an old age Jewish custom to write the abbreviations of ב”ה on a letterhead, or Sefer. The Rebbe encouraged this to be done on all letters and publications, even if they are of Torah matters. Some write the alternative of BS”D in order to avoid writing an abbreviation of Hashem’s name. Others avoid writing it all together.
Should one write B”H or BS”D on emails?
It is a good custom to do so. This especially applies to long letter emails, in which the custom of writing Beis Hei in its top corner is identical to the old age custom of writing it by a letter.
Should one write B”H or BS”D in text messages, chats, WhatsApp’s?
The custom is not to be particular to do so.
 See Sefer Chassidim 935 that one may erase two Yuds; Aruch Hashulchan Y.D. 276:28 that it has no Kedusha at all although may not be used for a belittling matter; Mishneh Halachos 6:183; See Piskeiy Teshuvos 154 footnote 121 in name of Mahariy Asad Y.D. 304; Maharsham 1:159 in Hahsmatos; Milameid Lehoil 113:20; Maharshag 1:3
 The reason: As some Poskim rule that even two letters of Hashem’s name does not have Kedusha, and certainly this would apply by a single letter even according to those who are stringent. [See Mishneh Halachos ibid]
 See Rama Y.D. 276:10 that it is only permitted to erase two Yud’s for a need; Shach Y.D. 276:14 in name of Terumos Hadeshen and Darkei Moshe that two Yud’s may only be erased for a great need; Chida in Bris Olam in name of Radbaz 207 regarding two Yuds; Maharsha Sukkah 53; Maharitz Chayos 1; Ruach Chaim [Falagi] Y.D. 276; Tzafnas Paneiach 196 “One is to beware not to erase it”; See Tzafnas Paneiach 303 that so is apparent from Rav Haiy Gaon, brought in Maggid Mishneh Shabbos 11:10 that even abbreviations are considered words regarding liability over writing on Shabbos; Igros Moshe ibid that even one letter of Hashem’s name contains sanctity
 Tzafnas Paneiach ibid “One is to beware not to write it on letterheads” [See Piskeiy Teshuvah 3:233 for the correspondence letters of the Imrei Emes with the Rogatchaver on this matter; However, see Migdal Oz p. 100 for a letter of the Rogatchaver to the Rebbe Rashab in which Beis Hei was written]; Igros Moshe Y.D. 2:138 [brought below]; Igros Kodesh 21:5 [letter 7759, brought in Shulchan Menachem 5:218 and Menachem Meishiv Nafshi 2:817] that the Rebbe asked the Rogatchaver Gaon as to the source of writing this, and in fact brought Talmudic proofs to discredit it, and that it is, and never was, customarily done in documents such as Kesubos, Gittin and loan documents. Interestingly, the Rebbe in two of his letters to the Rogatchaver indeed omitted the term B”H or any other opening term. However, in this letter he did write it, and so was the Rebbe’s final position as evident from other letters and from his custom.
Opinion of Igros Moshe: See Igros Moshe ibid that some avoid doing so being that it is considered part of Hashem’s name and it may come to be desecrated. The Igros Moshe negates this opinion being it is not common to reuse papers and they are simply burnt. Nonetheless, he concludes that he does not see any need to be particular to write B”H, being it is not mentioned in previous Poskim. Furthermore, why should Hashem’s name be mentioned on a paper that has secular matters, or even forbidden matters written on it, such as Lashon Hara.
 Igros Moshe ibid; Teshuvos Vehanhagos 1:640; Mishneh Halachos 6:183; Piskeiy Teshuvos ibid
 The following is a list of Geodlei Yisrael who did not write any heading on their letters: Teshuvos Vehanhagos 1:640 that the Gr”a and Rav Chaim Velozhin did not write any heading; Rogatachver; Chazon Ish, brought in Orchos Rabbeinu p. 247, would not even write BS”D; Igros Moshe; Rav Kapach; Rav Elyashiv; Rav Mashash
 Likkutei Sichos 6:190 “It is accustomed amongst all Jewry to write B”H in the beginning of the letter”; Mishneh Halachos ibid; Piskeiy Teshuvos ibid
 Letter of Imrei Emes [printed in Piskeiy Teshuvah ibid] in name of many Gedolei Yisrael; Orchos Rabbeinu ibid that so was the custom of the Steipler; So is evident from letters of the Sdei Chemed; Rav Ovadia Yosef; Rav Yoel Teitelbaum; Rav Wozner, and many other Gedolei Yisrael of today as can be readily verified in looking at their Haskama’s to Sefarim.
 See Sefer Maggid Dvrav Liyaakov in the Hosafos of letters in which the heading B”H is written on the letterhead of letters written by the Baal Shem Tov and many of his students, including: The Maggid; Pinchas of Koretz; Leib ben Sara’s; Reb Zusha of Anipoli; Shlomo Karlin; Rav Aaron Karlin; Rav Levi Yitzchak of Berditchiv; Rav Chaim Chaika of Amdura;
 See Igros Kodesh Admur Hazakein who writes B”H on all of his letters; See All letters of Rebbe Rashab, Rayatz and the Rebbe
 So can be readily seen from any Sefer Mamarim of the Rabbeim, including the Alter Rebbe [who writes BEZ”H] Mittler Rebbe [who interchanges between BEZ”H and BS”D], the Rebbe Rashab, Rebbe Rayatz, and Rebbe who would all write BS”D; See Hiskashrus 749 p. 9; Shulchan Menachem 5:220 footnote 4
 Igros Kodesh letter 1384 [brought in Menachem Meishiv Nafshi 2:650] “It is of upmost importance in today’s times to begin all relevant matters with the abbreviations of Beis Hei”; Likkutei Sichos 6:190 “It is the custom of all Israel, and a custom of Israel is Torah, that upon writing even a mundane letter, they begin to write Beis Hei and the like.”; 7:190; 24:599; Heichal Menachem 1:249; Letter to Rav Tzinner printed in Tzadik Limelech 7:230 and Menachem Meishiv Nafshi 2:755
 Letter to Rav Tzinner ibid the Rebbe wrote Beis Samech Daled; Likkutei Sichos ibid and ibid “and the like”
 Letter of Rebbe to Rogatchaver printed in Igros Kodesh 21:5
 See Maaneh of Rebbe in next footnote
 Igros Kodesh letter 1384 ibid “So too you should write it in the Torah publication, and if only that this would suffice.” Maaneh of Rebbe to a Kuntrus that he was handed “Thank you. It is a wonder that the abbreviations of Beis Hei and the like were omitted from the Kuntrus. We already have a Pesak Din from Moshe Rabbeinu that even a house filled with books requires a Mezuzah [and thus so too a Torah publication]. Even the Mamarim of the Rabbeim begin with a Beis Samech Daled. Now, the fact that in Likkutei Sichos this is not done, this itself is difficult [i.e. not to my satisfaction].”