We study the development and saturation of the m =1 one-armed spiral instability in remnants of binary neutron star mergers by means of high-resolution long-term numerical relativity simulations. Our results suggest that this instability is a generic outcome of neutron star mergers in astrophysically relevant configurations, including both "stiff" and "soft" nuclear equations of state. We find that, once seeded at merger, the m =1 mode saturates within ∼10 ms and persists over secular time scales. Gravitational waves emitted by the m =1 instability have a peak frequency around 1-2 kHz and, if detected, they could be used to constrain the equation of state of neutron stars. We construct hybrid waveforms spanning the entire Advanced LIGO band by combining our high-resolution numerical data with state-of-the-art effective-one-body waveforms including tidal effects. We use the complete hybrid waveforms to study the detectability of the one-armed spiral instability for both Advanced LIGO and the Einstein Telescope. We conclude that the one-armed spiral instability is not an efficient gravitational wave emitter. Even under very optimistic assumptions, Advanced LIGO will only be able to detect the one-armed instability up to ∼3 Mpc , which corresponds to an event rate of 10-7 yr-1 to 10-4 yr-1 . Third-generation detectors or better will likely be required to observe the one-armed instability.