<p>Flatband materials exhibit energy bands with zero dispersion, allowing wavefunctions to be compactly localized on the scale of a unit cell. However, such compact localization does not generally apply to wavefunctions with complex structures, such as those carrying orbital angular momentum (OAM). This limitation arises from the fact that a Bloch wavefunction consists of a plane wave factor <i>e</i>^{<i>i</i><b>k</b>⋅<b>r</b>} multiplied by a periodic function <i>u</i>(<b>r</b>). While a flatband flattens the dispersion of the plane wave factor, the compact localization of a general wavefunction additionally requires a highly degenerate periodic function to accommodate its internal structure. Here, we introduce a general framework for constructing such highly degenerate flatbands by leveraging bound states in the continuum (BICs). We experimentally demonstrate this framework in two- and three-dimensional (2D and 3D) acoustic crystals, realizing flatbands with four-fold and twelve-fold degeneracy, respectively. The resulting internal degrees of freedom enable the compact localization of complex structured fields with OAM in both 2D and 3D. Our results not only establish a viable platform for OAM-compatible flatband filtering for acoustic signal processing, but also open new avenues for the construction of topologically structured waves.</p>