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I see and hear lots of Rav Moshe's teshuvos quoted, mostly the "famous" ones.

For example, a number of notable Rav Moshe teshuvos can be found on Rav Moshe's Wikipedia page:

Does an English online compendium that collects (many/most/all) of Rav Moshe teshuvos exist?

As noted in the comments by several Mi Yodeyans ¹, the lack of a compendium/ collection of Rav Moshe's teshuvos translated to English is no coincidence: it's what Rav Moshe himself wanted.

Rav Moshe didn't want his teshuvos translated to English for fear that people would erroneously draw comparisons from his teshuvos and incorrectly apply his psakim.

As Rav Moshe writes Y.D. 4:38:

איסור לפרסם קיצורי דינים ופסקים מספרי תשובות וגם לתרגם תשובות ללשון המדינה

אף שאני כותב עתה תשובה ארוכה בדבר השמות אבל מכתבי זה הוא בקיצור בדבר ששמעתי שאחד עשה ספר בלע״ז בהעתקת דינים מהספר אגרות משה שלי, והוא איסור אף אם היה התרגום ראוי שבזמננו ליכא מי שיכול לומר ולהדפיס פסקים בלא באור ומקורות, וכבר בקש ממני רשות איזה אינשי מכאן ואמרתי שאיני נותן רשות על זה. וגם נמצא שלא העתיק כראוי ויש כמה דברים בטעותים וכמה דברים שגורמים לטעות שזה עוד גרוע ביותר, וגם להעתיק התשובות ממש פשוט שהוא חסרון גדול ליתן הפסקים לאינשי דעלמא שאינם ת״ח שיבואו לדמות מלתא למלתא ולכן אני מוחה בכל תוקף שיש למחות.

It is forbidden to publish abbreviations and rulings from responsa and to translate answers to the language of the state

... I heard that one of them had done a book in the Hebrew language in copying laws from the my sefer Igrot Moshe, which is a prohibition even if the translation was appropriate [...] a number of people asked me for permission [to translate Igrot Moshe into English] and I told them I did not give permission for this.

Rav Moshe also addresses this in Igrot Moshe Y.D. 3:91.

Note: only translated small portions of the teshuva since it would be not only ironic but wrong to go against Rav Moshe's wishes in the exact teshuva where he writes he doesn't want Igrot Moshe to be translated

(hopefully the 2 snippets I translated were ok- altz they can help teach and serve as a disclaimer to what Rav Moshe's preferences in this matter are)

^{1 _{@DoubleAA, @mbloch, @Alex thanks also for the 2 Igrot Moshe teshuvot referenced.}}

R. Feinstein's son in law R. Moshe Dovid Tendler published a translation/commentary of some of R. Feinstein's responsa relating to medical issues. The book is called Responsa of Rav Moshe Feinstein: Translation and Commentary Vol I Care of the Critically Ill. In his introduction R. Tendler discusses R. Feinsteins responsum (Y.D. 3:91) that prohibited the translation of his responsa, and R. Tendler cites a later responsum (Y.D. 4:38) to explain why his translation is okay. He also adds:

My great father-in-law, זצ"ל, in discussions at my home, where for decades he spent the Succos holy days, further elaborated on this point. He asked me to translate those responsa which I felt it was important for our brethren to understand, but to do so "as a rebbe teaches a talmid." He asked that I work with meticulous care, presenting all the relevant background material in a lucid style, with full awareness that the readership would span the spectrum from Torah scholars to people with minimal background. He added, as an aside, that an English translation of the responsa would be available to non-Jews as well. Although teaching Torah to non-Jews is prohibited, our purpose in translating these responsa is to instruct Jews, and therefore we need not consider this translation a violation of the edict concerning the study of Torah by non-Jews.

I have prepared the following translations in accordance with the above instructions. Although the English version is faithful to the original, I have included additional material, enclosed in brackets, to instruct and edify those who will turn to the writings of the greatest posek of our generation as to a personal rebbe.

Portions of this book are available online at Google Books, and it is available on Sefaria as well.

In Theology in the Responsa R. Louis Jacobs goes through each century of responsa literature and summarizes those responsa that relate to theology. In the section for the twentieth century he summarizes three of R. Feinstein's responsa. However, as far as I know this cannot be read online.