Contactless manipulation of microparticles using acoustic waves holds promise for applications ranging from cell sorting to three-dimensional (3D) printing and tissue engineering. However, the unique potential of acoustic trapping to be applied in biomedical settings remains largely untapped. In particular, the main advantage of acoustic trapping over optical trapping, namely the ability of sound to propagate through thick and opaque media, has not yet been exploited in full. Here we demonstrate experimentally the use of the recently developed technique of single-beam acoustical tweezers to trap microbubbles, an important class of biomedically relevant microparticles. We show that the region of vanishing pressure of a propagating vortex beam can confine a microbubble by forcing low-amplitude, nonspherical, shape oscillations, enabling its full 3D positioning. Our interpretation is validated by the absolute calibration of the acoustic trapping force and the direct spatial mapping of isolated bubble echos, for which both find excellent agreement with our theoretical model. Furthermore, we prove the stability of the trap through centimeter-thick layers of bio-mimicking, elastic materials. Finally, we demonstrate the simultaneous trapping of nanoparticle-loaded microbubbles and activation with an independent acoustic field to trigger the release of the nanoparticles. Overall, using exclusively acoustic powering to position and actuate microbubbles paves the way toward controlled delivery of drug payloads in confined, hard-to-reach locations, with potential in vivo applications.