No Dyes at all.
Your question is based on a misconception:
A dye is a coloured substance that chemically bonds to the substrate to which it is being applied, this distinguishes dyes from pigments which do not chemically bind to the material they colour.Wikipedia
The coloration does not chemically bond with the plastic in production. It is thus a pigment that is melted into the filament.
The range of pigments is ginormous. They start as simple as pure carbon black, range over natural occurring ones such as zinc white (ZnO) to products of chemistry like the dark blue copper phthalocyanine. Some pigments of these are toxic (lead white), others are also carcinogenic (chrome yellow), others are atop that radioactive (uranium trioxide, aka uranium yellow).
But what is actually used for coloring the filaments? Usually, manufacturers choose their pigments carefully based on 3 factors:
- thermally stable in the printing range
- as non-toxic as possible (to avoid needing to declare it on the MSDS)
- as cheap as possible to work with
Usually, this rules out all the highly toxic and radioactive ones, as that demands extreme caution to work with, upping prices.
Usually, we can't know. While the list of inorganic pigments is rather limited in some areas, it is by far not complete. It's better to take a look at the Forbes Pigment Database, which lists 11 categories of white pigments, 9 of them by chemical composition, one by origin and one for 3 samples of unknown composition. The 2 categories of violet pigments contain 20 different samples.
But why don't we know?
In most cases, the color of plastic does not bring any danger or changes the properties more significant than changing its melting point. As the chosen pigments are inert to most treatments, they don't need to be listed on the MSDS, and thus omitted, allowing the companies to keep them a trade secret that helps them compete against other companies for only they have this one specific color.
Our best indicator for what pigments are used thus is, if they use a trade name for a filament, such as "ultramarine blue", which however might not be in the filament at all.
When do we know?
In some cases, we might actually know what they put into a roll of filament based on odd characteristics.
For example phosphorescent green filament. It is most commonly made with either zinc sulfide or strontium aluminate, and we can rule out one or the other based on how bright they glow on the dark photo.
Another case in which we know what is in the filament is in the case of metal infused filaments, where it is part of the advertisement, that these filaments contain some amount of one metal powder or another. For example, the rusting filament contains iron. Then there is also filament that contains up to 80 % metal powder. Another similar bucket are wood- and stonefill filaments, where wood fibers or ceramics were added to give color and texture.