An example would be the spontaneous magnetization of a ferromagnetic material as a function of temperature. Scale-invariance gives rise to critical exponents through something called the renormalization group. But this intermediate attractor is a spacetime that has a naked singularity. Here we have an example where they occur for any kind of matter, and for any data in which we have fine-tuned just one parameter to get us to the black hole threshold. This is closer to violating cosmic censorship than any other example so far.
Gravitational waves have energy, and therefore they have mass.
There is some numerical evidence that even in this case we have critical phenomena at the black hole threshold. That means that they are a property of general relativity itself, and not of any particular kind of matter. By studying critical phenomena in gravitational collapse, we study important generic properties of very strong gravitational fields. Critical phenomena at the black hole threshold were discovered in numerical simulations - perhaps the nicest example where numerical work in general relativity has come up with something completely unforeseen. Now there is both numerical and analytical work going on.