Shirat HaYam

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Overview

Saying Shirat HaYam seems to be a minhag Eretz Yisraeli. It was then adopted, in the Geonim time, in Bavel although not everyone agreed on saying it. Some said it all the time, some said it only on Shabbat and some didn't say it at all.
During the Rishonim time, in Ashkenaz and France they said Shirat HaYam as part of Pesukei DeZimra although in Spain and Provence they said it after Yishtabach as not part of Pesukei DeZimra. Only later did everyone start saying Shirat HaYam before Yishtabach as part of Pesukei DeZimra.

Tannaim and Amoraim

Although Shirat HaYam is mentioned in the Bavli as being said on מנחה on Shabbat, there is no mention of it being said in Pesukei DeZimra. In the Tur and אבודרהם it says that we say Shirat HaYam since the Mekhilta is דורש the ט"ו תשבחות of Yishtabach from the verses of Shirat HaYam but it is not found in our Mekhilta.

Geonim

Saying Shirat HaYam as part of Pesukei DeZimra seems to have started in Eretz Yisrael.¹ In Bavel some people adopted the minhag and would say Shirat HaYam but only on Shabbat as found in תשובות ר' נטרונאי גאון but not everyone.² During the time of Rasag it seems that they started saying Shirat HaYam even on chol.

Rishonim

During the Rishonim time, most people said Shirat HaYam but they said at different parts of davening:

Ashkenaz and France - In Ashkenaz and France everyone said Shirat HaYam as part of Pesukei DeZimra before Yishtabach.
Spain and Provence - In Spain and Provence the minhag to say Shirat HaYam was less קבוע³ and was said after Yishtabach to show that it wasn't part of davening. Later on, in Spain they were מושפע from minhag Ashkenaz⁴ and they started saying Shirat HaYam before Yishtabach.