  Low-frequency vibrational modes play a central role in determining various
basic properties of glasses, yet their statistical and mechanical properties
are not fully understood. Using extensive numerical simulations of several
model glasses in three dimensions, we show that in systems of linear size $L$
sufficiently smaller than a crossover size $L_{D}$, the low-frequency tail of
the density of states follows $D(\omega)\!\sim\!\omega^4$ up to the vicinity of
the lowest Goldstone mode frequency. We find that the sample-to-sample
statistics of the minimal vibrational frequency in systems of size $L\!<\!L_D$
is Weibullian, with scaling exponents in excellent agreement with the
$\omega^4$ law. We further show that the lowest frequency modes are spatially
quasi-localized, and that their localization and associated quartic
anharmonicity are largely frequency-independent. The effect of preparation
protocols on the low-frequency modes is elucidated and a number of glassy
lengthscales are briefly discussed.
