  The existence of black holes and of spacetime singularities is a fundamental
issue in science. Despite this, observations supporting their existence are
scarce, and their interpretation unclear. We overview how strong a case for
black holes has been made in the last few decades, and how well observations
adjust to this paradigm. Unsurprisingly, we conclude that observational
evidence for black holes is impossible to come by. However, just like Popper's
black swan, alternatives can be ruled out or confirmed to exist with a single
observation. These observations are within reach. In the next few years and
decades, we will enter the era of precision gravitational-wave physics with
more sensitive detectors. Just as accelerators require larger and larger
energies to probe smaller and smaller scales, more sensitive gravitational-wave
detectors will be probing regions closer and closer to the horizon, potentially
reaching Planck scales and beyond. What may be there, lurking?
