Has LIGO detected any gravitational waves yet, or any hints of them? Is it just that LIGO is not sensitive enough or are we now entering disconfirmation territory? If we never detect gravitational waves with any instrument, what will this mean for our understanding of Relativity and Quantum Gravity?
LIGO hasn't detected any gravitational waves, but its sensivity means it would only detect waves in favourable circumstances so the failure to detect anything isn't a great surprise. LIGO is currently undergoing an upgrade to Advanced LIGO, and if this also fails to detect anything people will be starting to worry.
A quick Google will find you lots of articles on the performance of LIGO and Advanced LIGO. Google with "site:arxiv.org" for the definitive articles, or this SciAm article is a good summary.
I'm not sure what would happen if gravitational waves remain stubbornly undetectable. It would certainly cause some head scratching amongst general relativists. There is indirect evidence for gravitational waves from binary pulsars, so I suspect the initial reaction would be that our ideas for detecting them are at fault rather than that gravitational waves don't exist.