Focusing on the question of blowing during the silent amidah of musaf, R. E. Melamed has an in-depth article available here (Hebrew). I will summarize some of his key points in my answer here.
The original custom seems to have been not to blow during the silent amidah. This was certainly the case among the majority of the geonim, where the custom was that the silent amidah contained only seven (rather than nine) blessings.
Rambam (Hilchot Shofar, 3:10), Tur (Orach Chayyim 592) and Shulchan Aruch (Orach Chayyim 592:1) all describe the blowing as taking place during chazarat hashatz, and make no mention of blowing during the silent amidah.
However, Tur (loc.cit) does quote from Aruch:
והני דמחמירין תוקעין ל' כדיתבין ול' בלחש ול' על הסדר
Those that are stringent blow thirty when they sit [i.e. before musaf], thirty in the silent amidah and thirty with the order [of the blessings in chazarat hashatz].
The practice of blowing during the silent amidah was also adopted by those of a kabbalistic bent, notably Ari and his students. For instance, R. Chayyim Vital wrote in Sha'ar HaKavannot, Inyan Rosh HaShanah, D'rush 7
ול' תקיעות מעומד, במוסף בלחש, במלכויות זכרונות ושופרות.
And thirty blasts while standing, during the silent musaf, in malchuyot, zichronot and shofarot.
As with many kabbalistic practices, this was widely adopted by edot hamizrach and chassidim, which is why it is widespread in nusach sefard synagogues.
Nusach ashkenaz synagogues have by and large retained the traditional custom not to blow during the silent amidah. To quote Mishnah Berurah 592:1
יש מקומות שנוהגין לתקוע כשמתפללין בלחש אבל אין לנהוג כן לכתחלה שלא לבלבל המתפללים
There are those place that have the custom to blow when they pray silently, but ideally one should not do this, in order not to disturb those praying.
(For a different angle on the topic, R. M. Sternbuch in Mo'adim uZemanim I 7 has a nice presentation where he discusses the halachic pros and cons of each custom.)
