Nishmat

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Overview

The text of Nishmat seems to very early and parts of it are found both in the Yerushalmi and in the Bavli. But, saying Nishmat as part of Pesukei DeZimra is not found in the Bavli or Yerushalmi and it is unclear when they started saying it.
By the time of the Geonim, they were already saying Nishmat on Shabbat. But from the Rishonim it seems to be that Nishmat wasn't totally קבוע. Some people said it only after Yishtabach which indicates that it is not totally part of Pesukei DeZimra. On the other hand, some said Nishmat always also on chol.

Tannaim and Amoraim

Saying Nishmat on Pesach is mentioned in Pesachim 118a. There is a disagreement between R. Yehuda and R. Yochanan what "ברכת השיר" is:

מַאי בִּרְכַּת הַשִּׁיר? רַב יְהוּדָה אָמַר: ״יְהַלְלוּךָ ה׳ אֱלֹהֵינוּ״. וְרִבִּי יוֹחָנָן אָמַר: ״נִשְׁמַת כָּל חַי״.

Different sections from Nishmat are found in the Yerushalmi and Bavli containing parts of our Nishmat. In Yerushalmi Berakhot 1:5, there is a disagreement about what is the version of מודים דרבנן, Bar Kapara would say:

בר קפרא אמר לך כריעה לך כפיפה לך השתחויה לך בריכה כי לך תכרע כל ברך תשבע כל לשון לך ה׳ הגדולה והגבורה והתפארת והנצח וההוד כי כל בשמים ובארץ לך ה׳ הממלכה והמתנשא לכל לראש והעושר והכבוד מלפניך ואתה מושל בכל ובידך כח וגבורה ובידך לגדל ולחזק לכל ועתה אלהינו מודים אנחנו לך ומהללים לשם תפארתך בכל לב ובכל נפש משתחוים כל עצמותי תאמרנה ה׳ מי כמוך מציל עני מחזק ממנו ועני ואביון מגוזלו בא״י אל ההודאות.

In the Bavli (Berakhot 59b) we have a different section of Nishmat, which was R. Yochanan's version of ברכת הגשמים:

EN/HEע/E

על הגשמים כו׳.: ועל הגשמים הטוב והמטיב מברך והא״ר אבהו ואמרי לה במתניתא תנא מאימתי מברכין על הגשמים משיצא חתן לקראת כלה.

Although the text of Nishmat seems early it is unclear when they started saying Nishmat on Shabbat as part of Pesukei DeZimra.

Geonim

Nishmat is found in all of the siddurim of the Geonim. The נוסח is very similar to our נוסח and it only has slight differences.

Rishonim

Although Nishmat was said already in the Geonim time it still wasn't so קבוע in the time of the Rishonim. In ספר העיתים it says that they would say Nishmat after Yishtabach which shows that they didn't include Nishmat as part of the real Pesukei DeZimra. The Rambam also says "נהגו העם" which means that not everyone said it.
The רוקח says that there are חסידים who say Nishmat all the time not just on Shabbat.