When exporting from Maya there are a few things you need to make sure you have set up correctly. The first thing is to make sure that you have the Export skins box checked or the skinning information will not be exported and no animations will work on your mesh. The second is to make sure the Export Animations box is checked to ensure the animation data is also exported.
If you have Export only selected checked, you must be sure to select your mesh and skeleton nodes in your scene when exporting. If you are exporting selected and do not select the skeleton, it doesn’t matter if you have the export skins or export animations checked as there would be no underlying skeleton in the scene to deform the mesh.
Lastly, if you want to separate animation clips, you can do that with the animation groups options in the Babylon tools menu.
In this window, you can add an animation group and assign a frame range for the group. You can then address this named group in your code to play just that section of the timeline. If these suggestions don’t help unblock you, I agree with @bghgary that the fastest way to help is if you can provide your file, or a test file if you are unable to share your asset that has a simple skinned mesh that repros your issue we can help debug the process.