Psalms 3, 7, 10, 12, 13 (twice), 20, 22, 29, 33, 34, 40, 44, 49, 50, 52, 65, 66, 67, 68, 72, 77, 89 (twice), 94, 131, 137, 143 (twice), 146.
Are other ornaments more or less frequent? The shalshelet is the rarest. Others range from three to 100 times its frequency in the data I have so far. The next rarest is the pazer, at 82 occurrences in the psalms. This SHV interprets as a rise to the second and a descent to the reciting note by step within the mode. Then there is the illuy, at 154. She interprets it as a rise to the fifth (much as its similar sublinear sign is the fifth note of the scale). This one tends to the unmusical in some situations - see below Psalm 3:3 which contains two examples both taken at the fourth below. I begin to wonder if the ancients wanted to be able to reach any note without changing the reciting note. This would be a legitimate desire for extending a long harangue till the next down beat.
So back to the shalshelet - does its use reveal anything about what it ornaments? The occurrence in Amos is not an a particularly striking word, yet the ornament definitely draws attention to that word - and he said. Perhaps the 8 sayings on the extended reciting note are also pointing to a uniqueness in Amos and his saying.
The occurrences in the Torah are these, Genesis 19:16, on Lot's lingering, Genesis 24:12, on Laban's saying when he reaches the well, Genesis 39:8; on Joseph's refusal to his master's wife, and Leviticus 8:23, when Moses slew the ram at the consecration of Aaron and his sons.
The only way is to look at them in the Psalms is one by one and decide if the aural pattern surrounds something of significance that is not surrounded by other ornaments. That is a tough question. Thirty examples required. Do they tell us anything about this rare ornament. And if they did, what authority do the ornaments carry? Some consider them late, but Mitchell here argues for a second to third century BCE manuscript tradition.
רַבִּים֮ אֹמְרִ֪ים לְנַ֫פְשִׁ֥י אֵ֤ין יְֽשׁוּעָ֓תָה לּ֬וֹ בֵֽאלֹהִ֬ים סֶֽלָה׃
7:6 יִֽרַדֹּ֥ף אוֹיֵ֨ב ׀ נַפְשִׁ֡י וְיַשֵּׂ֗ג וְיִרְמֹ֣ס לָאָ֣רֶץ חַיָּ֑י וּכְבוֹדִ֓י ׀ לֶעָפָ֖ר יַשְׁכֵּ֣ן סֶֽלָה׃
10:2 בְּגַאֲוַ֣ת רָ֭שָׁע יִדְלַ֣ק עָנִ֑י יִתָּפְשׂ֓וּ ׀ בִּמְזִמּ֖וֹת ז֣וּ חָשָֽׁבוּ׃