According to Wikipedia and validated by a clever thought experiment here, sound waves can be transverse as well as longitudinal, if they're propagating through a solid. Consider my mind blown and my curiosity piqued. However, is this a phenomenon we can hear? And is there any reason these waves would be different from longitudinal sound waves?
Understandably, this kind of sound wouldn't travel through the normal path as that requires traveling through the air to get to the eardrum. Bone conduction headphones seem to offer a promising avenue, but I see no reason that those sound waves wouldn't be longitudinal.
sound transmission through a solid can occur by either compressive waves or shear (transverse) waves because a solid is capable of sustaining shear stresses. sound transmission through air is exclusively by compression waves because air cannot sustain shear stresses.
You can certainly hear both sorts of waves by pressing your ear against, for example, a steel girder which is carrying both sorts of waves because someone on the other end of the girder is whacking it with a hammer, but the amount of each you will hear depends on details of how exactly your ear is coupled to the girder.