Mizmorei Shabbat

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Overview

Adding special Mizmorim on Shabbat is a later minhag and is not found in ספרות התנאים ואמוראים. It's found first in siddurim of the Geonim which mention saying "מזמור שיר ליום השבת" in Pesukei DeZimra.
During the Rishonim time more mizmorim were added to Pesukei DeZimra. In Ashkenaz and France they added Tehillim 19, 34, 90-91, 135-136, 33 and 92-93. In Spain and Provence it took longer until the mizmorim became קבוע. In the beginning everyone added any mizmorim, some said הלל הגדול and some said שיר המעלות. Later on, they adopted minhag Ashkenaz and started saying the same mizmorim as they said (in the order of Tehillim) in addition to a couple שירי המעלות and Tehillim 98.
Not all of these mizmorim were said only on Shabbat. According to סידור ראב"ן they were said always and only because everyone was busy working on chol were taken out of tefillot chol and were said only on Shabbat. But, it could have been also the opposite, the mizmorim were added on Shabbat and then since they were just regular Tehillim some people started saying them always in Pesukei DeZimra.

Tannaim and Amoraim

Adding special mizmorim to Pesukei DeZimra on Shabbat is not found in Tannaim and Amoraim literature and seems to be a later minhag.

Geonim

In the siddurim of the Geonim¹ the only mizmor which is found is Tehillim 92 "מזמור שיר ליום השבת". The rest of the mizmorim were only added later.

Rishonim

During the Rishonim time everyone said מזמור שיר ליום השבת, but not everyone added the same mizmorim.

Ashkenaz and France - In Ashkenaz and France everyone said the same mizmorim: Tehillim 19, 34, 90-91, 135-136, 33 and 92-93. In סידור ראב"ן it says that in the beginning they would say all the mizmorim on chol but since everyone were busy working on chol they are said only on Shabbat.
Spain and Provence - In Spain and Provence the mizmorim were less קבוע and they were said before Barukh Sheamar as a sign that it wasn't really part of Pesukei DeZimra. In ספר העיתים it says one says מזמור שיר ליום השבת and then one could add any mizmorim one wants. The Rambam says that some say הלל הגדול (Tehillim 135-136) and some say שיר המעלות.
Only later as found in the Avudraham² were the mizmorim that were said in Ashkenaz and France added also in Spain, but in Spain they said the mizmorim in the order of Tehillim (Tehillim 19, 33-34, 90-91, 98, 121, 124 and 135-136).