[cynest]

We have significant evidence to believe that gravitational waves do exist. By measuring the energy of Hulse-Taylor Binaries, we can show that the expected energy loss is there. This was the basis for the 1993 Nobel in physics.

[PurpleHouse]

I'd say more than significant evidence. They're predicted by General Relativity. If they don't exist, it would mean our understanding of the universe is seriously flawed. Not many theories of gravity can exist without gravity waves. I think the only one people take seriously is string theory. Though, string theory can exist with or without gravitational waves as it's one of the 'tunable' parameters.