My graduate research was on a corner of the class of ceramic materials that are used as coatings on top of this type of turbine blades. Anything specifically you would like to know?

Generally, single crystal blades are designed for their resistance to creep at high temperatures. If your predominate creep mechanism is sliding at grain boundaries in a polycrystalline material, you can impart creep resistance by removing the grain boundaries. A single crystal is the limiting case of this process.

3D printed parts, so far, are polycrystalline, and would be used in other places in an engine's hot stage.

Happy to address specifics.

What sorts of considerations are important for the ceramic coating? I'd assume that the coating is not under the same types of stress as the alloy core, and so thermal properties would be more important. Is that true, or is load as much of a factor there too?

I'm not deeply aware of the specifics of the mechanical loads near the surface due to rotation. Generally the mandates are to match CTE between the substrate and surface as much as feasible, and to insert a "bond" layer to help that match.

The dominant concerns I am aware of are low thermal conductivity, thermal stability, and lack of chemical attack in the environments in question.

Perhaps late to the thread, but turbine blades with internal cooling channels are one of the dream applications for metal additive manufacturing, though I'm not sure it may ever be realizable. If you could lower the operating temperature of the turbine blade in this way, you might be able to mitigate the risk of creep, which as sibling posters point out is the motivation for the use of single-crystal turbine blades and thermal barrier coatings.

There's a lot of buzz in the materials community around additive, but I think 3D-printed turbine blades are basically just a dream right now. People are still wrestling with getting consistent build quality for simple structural components...

I can see how 3D printing can be used for parts with complex shapes, as a replacement for powder sintering, but there is no way you can use it to cast a single crystal, neither for any part where you need to control texture.