Majorana zero modes (MZMs) are highly sought-after states with a possible application in quantum computation. Here, we show that vortex-core MZMs can carry a nontrivial angular momentum. This establishes distinct ‘‘flavors’’ of Majorana modes, independent of the Chern classification of topological superconductors. The MZM angular momentum is explicitly calculated for a microscopic model of a 𝑑+𝑖⁢𝑑 superconductor placed on a three-dimensional topological insulator (the 𝑑+𝑖⁢𝑑+Dirac model) using both exact diagonalization and the Chebyshev expansion. We classify all possible quantum numbers of MZMs depending on the windings of the order parameter and underlying normal state. The topological protection of the MZM is set by the bulk gap, quasiparticle poisoning by trivial in-gap states, and its localization length. All these severely limit the stability of MZMs in the 𝑑+𝑖⁢𝑑+Dirac model, in contrast to earlier claims. Nevertheless, the possibility of having different flavors of MZM—in the form of angular momentum or something else—can provide a unique path forward for the study of MZMs.