Since no one has listed a summary of concise rulings, I will list R. Bleich's conclusion from his Survey of Recent Halakhic Periodical Literature. I have inserted the sources that he mentioned throughout the survey in brackets.
The silk screen method is certainly subject to challenge on the basis of its inherent incompatibility with a number of considerations:
- The position of Bah [O''H 32:15], who maintains that each word must be vocalized before writing even when there is no chance of scribal
error;
- The view of the authorities who maintain that each word must be copied from an existing text and that an error-free “master text” is
not sufficient for this purpose [Bnei Yonah no. 271, Pilpul Arukh, p.
14a; Piskei Uzi'el, no. 31]
The view of the authorities who maintain that multiple Divine Names cannot be sanctified even simultaneously [Bnei Yonah 276:2; Divrei
Hamudot, Halakhot Ketanot, Hilkhot Sefer Torah 4:11; Ma’adanei Tom
Tov, Halakhot Ketanot, Hilkhot Sefer Torah 4:5; Eliyahu Rabbah 32:36;
Teshuvot Bet Shlomoh, Yoreh De’ah, II, no. 163; Teshuvot Dvar Shmu’el,
no. 76; Melekhet Shamayim, no. 5; Birkei Yosef Yoreh De’ah 276:3; Bet
Aharon, no. 12; Teshuvot Maharam Shik, Yoreh De’ah, no. 276; Kol
Ya’akov, Yoreh De’ah 274:4; R. Yitzchak Dov Bamberger, Melekhet
Shamayim, Binah 9:3; and Hazon Ish, Orah Hayyim 6:13, Yehaveh Da'at,
VI, no. 57; ; Teshuvot Helek Levi O''H, no. 26]
Iggeret Mosheh’s view [O''H, IV, no. 40 sec. 10] that any method in which sirtut serves no purpose is disqualified;
- The view of Teshuvot Zera Avraham [Y''D 117], She’ilat David [no. 7 sec. 2], Da’at Kohen [no. 160] and Piskei Uzi’el [no. 31] that the
printing press is not acceptable because “it is not in the manner o f
writing.”
The kabbalistic view expressed by Ari [Bnei Yonah, Pilpul Arukh, no. 271, p. 15a] regarding the particular form of writing that governs
the manner in which the writing of each letter is begun.
Nevertheless, a competent rabbinic decisor might, with justification, conclude that those are minority views and consequently rely upon the weight of authority in ignoring such concerns. The crucial consideration is whether or not the silk screen method
constitutes either “spilling” or “dripping” and hence is ruled invalid
by the Palestinian Talmud [Gittin 2:3 and Shabbat 12:4]. Since the
validity of Torah scrolls, tefillin and mezuzot is a matter of
biblical law, any doubt, if the there is indeed such doubt, must be
resolved in the negative. Thus, it would be necessary either to adduce
strong precedent in support of the validity of a method similar to the
silk screen process or to advance a compelling logical distinction
between that method and the processes described by the Palestinian
Talmud. In this writers opinion, the requisite demonstration has not
been forthcoming.
In addition, R. Bleich notes the reaction of contemporary rabbinic authorities, who have overwhelmingly prohibited the silk screen method and noted that Torah scrolls and ST'aM that were silk screened are unfit for use. In the form of letters signed and occasional responsa released, which were published in Yated Ne’eman, 24 Kislev 5763; 16 Kislev 5763; Yated Ne’eman, Parashat Mikez 5763, and an earlier letter dated Tammuz 5762.
The signatories prohibiting silk screen include: R. Ovadia Yosef, R. Yosef Shalom Eliashiv, R Samuel ha-Levi Woszner, R. Nissim Karelitz, the members of the Bet Din of Jerusalem’s Edah ha-Haredit, R. Chaim Pinchas Scheinberg, and R Menachem Yehudah ha-Levi Gross [who wrote a detailed analysis explaining the prohibition in the Tammuz 5762 issue of Or Torah]. You can read a translation R. Wosner's letter here.
Besides R. Abadi's responsum [Or Yizhak, no. 53], the only published source that somewhat defends the silk screening process is an anonymous pamphlet called Ha-Emet ve-ha-Shalom Ahevu, who ultimately concedes that it shouldn't be done as R. Eliashiv has ruled against it.
UPDATE:
R. Abadi has responded to the letters of Gedolim who prohibited silk screening, arguing that they didn't see the process, and claims that "Rav Scheinberg, Rav Tuvia Goldstein ZT"L who both saw the process and Rav Dovid Feinstein, who did not see it, among "others", said it is Kosher." R. Abadi has also responded to R. Bleich's survey, in which he commends the thorough treatment and respect in the article. He notes that the "crucial consideration" of spilling does not happen in silk screening, and invites anyone to see the process to prove his case.
Since R. Bleich's article (published in 2003), several responsa have come out. As sam pointed out, R. Belsky has prohibited the practice in his Sha'alos U'Teshuvos Shulchan Halevi. R. Ovadia has also written about the grave sin of the silk screen method in Chazon Ovadia Purim (רנה), saying that nobody who uses a Torah scroll or ST'aM that was silk screened fulfills any mitzvah. The newest Yalkut Yosef, in 691:15, cites Chazon Ovadia in prohibiting the practice, adding that not only are they passul b'dieved, but also:
והמברכים עליהם ברכתם ברכה לבטלה, והוא מכשול חמור מאד, שמכשיל את הרבים באיסור ברכה לבטלה ובביטול מצות מקרא מגילה.
One who blesses on them [megillah] has said a blessing in vain, and this is a very serious stumbling block, thwarting the public through the prohibition of saying a blessing in vain and wasting the mitzvah of reading the megillah.