The existence of black holes and spacetime singularities is a fundamental issue in science. Despite this, observations supporting their existence are scarce, and their interpretation is unclear. In this Perspective we outline the case for black holes that has been made over the past few decades, and provide an overview of how well observations adjust to this paradigm. Unsurprisingly, we conclude that observational proof for black holes is, by definition, impossible to obtain. 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 coming years and decades, we will enter an era of precision gravitational-wave physics with more sensitive detectors. Just as accelerators have required larger and larger energies to probe smaller and smaller scales, more sensitive gravitational- wave detectors will probe regions closer and closer to the horizon, potentially reaching Planck scales and beyond. What may be there, lurking?