  Phonon trapping has an immense impact in many areas of science and
technology, from the antennas of interferometric gravitational wave detectors
to chip-scale quantum micro- and nano-mechanical oscillators. It usually relies
on the mechanical suspension--an approach, while isolating selected vibrational
modes, leads to serious drawbacks for interrogation of the trapped phonons,
including limited heat capacity and excess noises via measurements. To
circumvent these constraints, we realize a new paradigm of phonon trapping
using mechanical bound states in the continuum (BICs) with topological features
and conducted an in-depth characterization of the mechanical losses both at
room and cryogenic temperatures. Our findings of mechanical BICs combining the
microwave frequency and macroscopic size unveil a unique platform for realizing
mechanical oscillators in both classical and quantum regimes. The paradigm of
mechanical BICs might lead to unprecedented sensing modalities for applications
such as rare-event searches and the exploration of the foundations of quantum
mechanics in unreached parameter spaces.
