Shulchan Aruch Chapter 592:
The Loud Mussaf Service and the Order of the Shofer Blows (4 Chapters)
1. The Chazzan repeats the prayer and we blow the Shofer according to the order of the blessings. For מלכיות (blessings of Kingship), Tekiah-ShevarimTeruah-Tekiah one time. For זכרונות (blessings of Rememberance), Tekiah-Shevarim-Teruah and for שׁופרות (blessings relating to the Shofer), Tekiah-Teruah-Tekiah. But nowadays the custom is to blow for מלכיות (blessings of Kingship), Tekiah-ShevarimTeruah-Tekiah three times. For זכרונות (blessings of Rememberance), Tekiah-Shevarim-Teruah three times and for שׁופרות (blessings relating to the Shofer), Tekiah-Teruah-Tekiah three times.
[Some say that we plow Tekiah-ShevarimTeruah-Tekiah for Kingship blessings one time and the same is for Remembarance blessings and for the Shofer blessings. Similarly, the custom in these countries and our custom is to say היום הרת עולם (today is the birthday of the world...) and ארשׁת (may the music of our mouths...) every time after they blow. Even on the Sabbath where we do not blow Shofer we say היום הרת עולם but not ארשׁת.]
2. An individual does not interrupt his Shofer blowing with blessings even if he has someone to blow for him. [Rather, they blow for him before he says the Mussuf prayer and it is not necessary to blow a second time for him]
3. There may be no talking either for the blower or the congregation. This applies to both the separate Shofer blows (תקיעות שׁמיושׁב) and those said in the order of blessings (תקיות שׁמעומד). [However, that needed for the blowing and prayer is not an interruption.] And if one talked about irrelevant matters, one need not repeat the blessing and do it over.] And I need not say that one may not talk between the blessing and blowing if it is not necessary for the blowing.
4. The one blowing for the separate Shofer blast should blow for the Shofer blasts said in the order of blessings. [However, this is not inhibitory and someone else may do the blowing. But it is proper to do it as such and I already wrote about this topic in 585.]