I was surprised to see the Mishnah Brura on Orach Chaim Siman 124:4 says that not only must one be quiet during Chazarat Hashatz, one must not sing with the Baal Tefillah. How do we reconcile this with the widespread practice of singing with the Chazan (e.g. Sim Shalom) or saying various phrases with him (e.g. Tiferes Gedulah/ Avrohom Yagel during Mincha Shabbat)? Is this a problem?

The Mishna Brura (ibid 16) mentions two possible problems. The first – if a person says the Bracha with the Chazzan he might say Hashem's name and a Bracha in vain. The second problem might be if a person is singing out loud during the repetition, boasting with their beautiful voice, which is also inappropriate. But if as part of Tfilah all the congregation is singing along like during the Halel or any other part of Tfilah – that is permissible.