What is known regarding which hyperbolic groups are cubulated?

I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.

My impression is that not all of them are, but I didn't manage to find references with a counterexample.

Are there known ways to create non-cubulated hyperbolic groups? Are there famous examples of such?

What is known regarding which hyperbolic groups are cubulated?

I take it the usual definition of cubulated is acting properly and cocompactly on a CAT(0)-cube complex.

My impression is that not all of them are, but I didn't manage to find references with a counterexample.

Are there known ways to create non-cubulated hyperbolic groups? Are there famous examples of non-cubulated groups?