This runs by default on the CDTV. In essence, if something works on an Amiga 500, the CDTV might be able to run it. Of course, it's probably not as simple as this…

https://www.youtube.com/watch?v=nQ4gRTyR7BQ&t=980s

https://www.gregdonner.org/workbench/wb_cdtv.html

https://cloanto.com/amiga/roms/

on the CDTV, and while it is possible (with effort) to make 2.x and 3.x work

From io55's searching, this appears to be the only mention online of AmigaOS 2.x or 3.x on the CDTV. Experimentation could be warranted here.

https://www.hyperion-entertainment.com/index.php/frequently-asked-questions/55-amigaos-314/191-why-were-the-amiga-cdtv-and-the-amiga-cd32-left-out-of-this-amigaos-update

This device runs an operating system which is known to emulate various devices with EOPs. See the AmigaOS page.

Running on a variety of Amiga models, Minix can also run on the CDTV.

https://www.youtube.com/watch?v=bNKJD7fCa9E

[…]MMU requirement; it will not run on[…]CDTV[…]out of the box[…]must install a CPU board

With the installation of a CPU board to add an MMU, NetBSD can run on the CDTV. Of course, the CDTV at this point will be closer to an Amiga than it's stock state.

http://wiki.netbsd.org/ports/amiga/